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Beginning analysis

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The following are from the course notes by Aaron Tikuisis1. I really enjoy Prof. Tikuisis’s lectures in analysis. He brings a particular level of clarity to developing the intuition behind a lot of the less translucent definitions and theorems in introductory analysis.

The purpose of this post is to hopefully aid other people in their studies. I find it helpful to have the definitions, remarks, corollaries etc. listed out somewhere for reference away from the examples and exercises. Maybe some will find this useful as well.

Hovering over a definition, lemma, corollary, theorem, etc. reveals a sublink that displays only that specific section. This can be a helpful, non-distracting way to reference material. Enjoy.

Real Numbers

Definition 1.1.1

A field is a set equipped with two operations, addition , and multiplication, , which satisfy the following properties:

(F1) For all , we have
(F2) For all , we have
(F3) There is an element such that for all
(F4) For any , there is an element such that
(F5) For all , we have
(F6) For all , we have
(F7) There is an element such that and for all
(F8) For any , there is an element such that
(F9) For any ,

Definition 1.2.1

An ordered field is a field equipped with a binary relation satisfying the following properties

(O1) If and then
(O2) For all , exactly one of the following hold: or or
(O3) For all if then.
(O4) For all if and then

There’s some other facts, that the real numbers is an ordered field. So is but is not2.

Definition 1.3.1

Let be an ordered field, let be a subset, and let . Then is an upper bound for if for any ,

Likewise, is a lower bound for if for any ,

is bounded above if there exists an upper bound for . is bounded below if there exists a lower bound for . is a bounded set if it is bounded above and bounded below.

Note that sometimes, a set is bounded below by an element in that set: is bounded below by , which is in . We say that is the minimum of and write . Likewise, if a set contains an upper bound , then is the maximum of and write .

However, some bounded sets don’t contain their bounds. For example doesn’t contain any element which is a lower bound for .

Do note that whenever you write or , you imply that .

Definition 1.3.5

Let be an ordered field, let be a nonempty subset, and let . is a least upper bound (or supremum) for if:

(i) it is an upper bound for , and
(ii) for any upper bound of , we have .

When is a least upper bound for , we write

Similarly, is a greatest lower bound (or infimum) for , written , if

(i) it is a lower bound for , and (ii) for any lower bound of , we have .

By writing or , we suggest that the supremum and infimum of a set (if they exist) are unique.

We also have the following conventions. If is not bounded above, we write . If is not bounded below, we write . We also write and .

Definition 1.3.8

An ordered field is complete if for any nonempty set which is bounded above, there is an element which is a least upper bound for . In short, if is nonempty and bounded above, then exists.

A fundamental fact is that the real numbers is complete3.

Proposition 1.3.11

For any set which is nonempty and bounded below, exists.

Proposition 1.3.12

There exists a real number such that .

Theorem 1.3.13 (The Archimedean Property)

The set is not bounded above4.

Proposition 1.4.1

The absolute value of a real number is defined by

(i)
(ii)
(iii)
(iv) by the triangle inequality
(v) by the reverse triangle inequality

Definition 1.4.2

Let . The distance between and is

Proposition 1.4.3

Let . Then

(i)
(ii)
(iii)

Sequences

Definition 2.1.1

A sequence of real numbers is a function .

The definition above is a formal definition of a sequence. However, in practice, we think of a sequence as an infinite list of numbers. The conventional notation for a sequence is or . In both cases, the sequence corresponds to the function by .

Definition 2.1.2

A sequence is bounded if the set is bounded. Likewise, we may define to be bounded above or bounded below, by the corresponding property for .

We want to emphasize sequences are not sets. Order in a sequence matters. Repeats in a sequence matter.

One of the primary things we’re interested in with sequences is what looks like when is large.

Definition 2.2.1

Let be a sequence of real numbers and let . We say that converges to if for every there exists such that for all ,

In this case we may write

If a sequence does not converge to any real number, then we say it diverges.

Proposition 2.2.4 (Uniqueness of limits)

Let be a sequence and let . If

Then .

Definition 2.2.5

Let be a sequence of real numbers. We say that diverges to if for every there exists such that for all ,

Likewise we say that diverges to if for every there exists such that for all ,

If diverges to we may write

If it diverges to we may write

Proposition 2.2.8

Let be a sequence which converges to some number . Then is bounded.

Proposition 2.3.1 (Algebra of limits)

Let and be sequences converging to and respectively, and let . Then

(i) converges to
(ii) converges to
(iii) converges to
(iv) If for all and then converges to .

Remark 2.3.2

(i) and (ii) of the above proposition say that the set of converging sequences forms a vector space, and that the map is a linear map from to .

Proposition 2.3.3

Let and be converging sequences. If

Then

Corollary 2.3.4

Let be a converging sequence such that

Then

Theorem 2.3.5 (Squeeze Theorem)

Let , , be sequences such that:

(i) and converge to the same number , and
(ii) for all

Then also converges to .

Proposition 2.3.7

Let and be sequences such that

Then

(i) If then
(ii) If then

Proposition 2.3.9

For any real number , there exists a sequence of rational numbers5 (i.e., for all ) such that

Now, we move on to sequences which are monotone, meaning that they always tend in the same direction either increasing or decreasing.

Definition 2.4.1

Let be a sequence.

(i) We say is (weakly) increasing if

(ii) We say is strictly increasing if

(iii) We say is (weakly) decreasing if

(iv) We say is strictly decreasing if

(v) Finally we say is monotone if it is either increasing or decreasing6.

Theorem 2.4.2 (Monotone Convergence Criterion)

Let be a monotone sequence. Then it converges if and only if it it is bounded.

Specifically, if is bounded and increasing, then it converges to , whereas if it is bounded and decreasing, then it diverges to .

Now given a sequence ,

We may want to forget that parts of the sequence and only take certain terms. What we get when we do is called a subsequence.

Definition 2.5.1

Let be a sequence. A subsequence is a sequence of form , where is a strictly increasing sequence with for all .

Note that we index a subsequence with a different variable ( instead of ). Since the subsequence arises from the sequence, we typically don’t write for a subsequence.

We require the indices of the subsequence, , to be strictly increasing. So a sequence with repeating indices is not a subsequence.

Proposition 2.5.4

If converges to , then any subsequence also converges to .

Proposition 2.5.5

Every sequence contains a monotone subsequence.

Here begins an important corollary of the previous proposition.

Corollary 2.5.6 (Bolzano-Weierstrass Theorem)

Every bounded sequence has a convergent subsequence7.

It is especially useful to characterize when a sequence converges in a way that doesn’t require knowing what the sequence converges to.

Definition 2.6.1

A sequence is Cauchy if for all there exists such that for all ,

Theorem 2.6.2 (Cauchy Convergence Criterion)

Let be a sequence of real numbers. Then it converges if and only if it is Cauchy.

Definition 2.7.1

Let be a sequence of real numbers. The limit superior of is

The limit inferior of is

In other words, we can characterize the limi superior as the infimum of the set of eventual upper bounds for , and similarly for the limit inferior. Using the notational conventions previously, we write the following:

  • If is not bounded above then (since we have no eventual upper bounds),
  • If is not bounded below then (since we have no eventual lower bounds),
  • If diverges to then (since every number, no matter how negative, will be an eventual upper bound),
  • If diverges to then (since every number, no matter how large, will be an eventual lower bound),

Proposition 2.7.3

For any sequence ,

We now give another formula for and , which justifies the notation.

Proposition 2.7.4

Let be a bounded sequence of real numbers. Then

Remark 2.7.5

In the proof for Proposition 2.7.4 if we assume only that is bounded above, then the are still real numbers, and the same arguments will still work (although there exists the possibility that the limit of will be negative infinity and is equal to that). In the case that is not bounded above, this means that for all . However, if we agree that , then under this convention, again holds. So adopting the right conventions, is bounded is not necessarily required in the above proposition.

Theorem 2.7.6

Let be a sequence of real numbers. Then converges if and only if and this value is finite. In this case,

Series

We now move on to series (of real numbers). This is an infinite sum where is a sequence of real numbers. Since addition in is only defined for finitely many numbers at a time8 we need to define what an infinite sum means.

Definition 3.1.1

Let be a sequence of real numbers. For define

called the th partial sum of the series . We say that the series converges (to ) if the sequence converges (to ), and in this case, we may also write

If a series does not converge, we say it diverges. If diverges to , we write

In other words,

provided that the limit on the right exists.

Proposition 3.1.2

Let and be sequences and let . Suppose that the series

both converge.

(i) converges, and

(ii) converges, and

The above says that the set

is a vector space and the function defined by is a linear map.

Proposition 3.1.3

Let and be sequences such that for all . If the series

both converge, then

Proposition 3.1.4

Let be a sequence of real numbers and let . Then

converges if and only if

converges.

Example 3.1.5

Consider the series

Note that

Therefore, we may simplify the partial sum

by cancelling terms in the middle. Therefore,

A series in which the partial sums simplify in this way is called a telescoping series.

Typically, series are not telescoping, so we want more systematic techniques for determining whether a series converges.

Example 3.1.6

The harmonic series is

Consider the partial sum

We have terms which are , so their sum is at least . Following this, we have terms which are , so their sum is at least . Continuing, we can see that each grouping sums to at least . There are such groupings, so we have

It follows that is not bounded above, so it diverges. In fact, the sequence is increasing, so we see that and thus

The previous example is very important, since . Intuitively, one might have expected that a series converges if and only if its terms converge to ; while the harmonic series shows that one direction of this is not true, the intuition is correct in the other direction, as the next result shows.

Proposition 3.1.7

Let be a sequence of real numbers. If the series

converges, then

Example 3.1.8

A geometric series is one of the form

where . When , the sequence does not converge to , so by the Divergence Test, it follows that the corresponding geometric series diverges.

On the other hand, for , the sequence does converge to , so the corresponding geometric series at least has a chance to converge. We rewrite the partial sum

From this we see that the series does converge,

Proposition 3.2.1 (Boundedness Test)

Let be a sequence of real numbers. Suppose that:

(i) for all , and

(ii) There is a bound on the partial sums, so that

for all .

Then converges, and in fact .

Proposition 3.2.2 (Comparison Test)

Let and be sequences such that

Then:

(i) if converges, then so does .

(ii) if diverges, then so does .

Warning: it is easy to get confused about the hypothesis of the Comparison Test. If , and converges, then we cannot conclude anything about . Likewise, if diverges, we cannot conclude anything about .

Proposition 3.2.3 (Absolute Convergence Test)

Let be a sequence of real numbers. If the series

converges, then so does the series

Example 3.2.4

One calls a series absolutely convergent when the series converges.

It is a nontrivial result (by Dirichlet and Riemann) that a series is absolutely convergent if and only if any rearrangement of it converges to the same value, i.e.,

for any bijection 9.

As an illustration of this, consider the alternating harmonic series

This series converges to some value (by the Alternating Series Test, Proposition 3.2.9 below; the estimate comes by looking at the second partial sum). If rearrangements were allowed, then we could do the following

Since , this is a contradiction. This shows that rearranging the alternating harmonic series can change its value, confirming a special case of the Dirichlet–Riemann result, since we already know that the alternating harmonic series does not converge absolutely, by Example 3.1.6.

Proposition 3.2.5 (Ratio Test)

Let be a sequence of nonzero real numbers.

(i) If

then converges (absolutely).

(ii) If

then diverges.

Proposition 3.2.7 (Root Test)

Let be a sequence of real numbers.

(i) If

then converges (absolutely).

(ii) If

then diverges.

Remark 3.2.8

Although will often converge in examples you will see, it might converge to , in which case the Root Test tells us nothing. Again this can be seen through the examples

However, in both cases, the converges to .

Proposition 3.2.9 (Alternating Series Test)

Let be a sequence of real numbers. Suppose that:

(i) is a decreasing sequence, and

(ii) .

Then

converges. Moreover, for any ,

Proposition 3.2.10 (Integral Test)

Let be a function. Suppose that:

(i) for all , and

(ii) is decreasing: whenever .

Then the series

converges if and only if the improper integral

converges.

Proposition 3.3.1 (Cauchy Convergence Criterion)

Let be a sequence of real numbers. Then converges if and only if, for every there exists such that, for all ,

Topological Concepts

Definition 4.1.1

Let be a set.

(i) is open if for every , there exists such that the open interval is contained in .

(ii) is closed if its complement,

is open.

It is not the case that a set is either open or closed10. For example, the set is neither open (since but is not a subset of for any ) nor closed (similar reasoning with ).

Example 4.1.2

Any open interval is an open set.

First consider a bounded open interval . Given we have and , so we may set

Then we see that

and so .

For an unbounded open interval or , similar reasoning applies. Likewise, unbounded closed interval or is closed, since each is the complement of an unbounded open interval ( or respectively).

Proposition 4.1.3 (Permanence properties of open sets)

(i) The sets and are open.

(ii) For any finite collection of open sets , their intersection,

is open.

(iii) For any arbitrary collection of open sets , their union,

is open.

Example 4.1.4

Although we have that finite intersections of open sets are open, it is not true that infinite intersections of open sets are open. For example,

but isn’t an open set because it contains no open interval centred at .

Example 4.1.5

For any , the bounded closed interval is closed, since

which is open by the previous example and since unions of open sets are open.

Proposition 4.1.6 (Sequential characterization of closedness)

Let . Then is closed if and only if, for every sequence in (i.e., for all ), if converges then

Definition 4.1.7

Let be a set and let .

(i) is an interior point of if there exists such that . The interior of is

(ii) is an accumulation point of if there is a sequence in such that . The closure of is

(iii) is a boundary point of if it is an accumulation point of and it is not an interior point. The boundary of is

(iv) is an isolated point of if there exists such that .

(v) is a limit point of if it is an accumulation point and it is not an isolated point of .

Example 4.1.8

Let .

  • The interior points of are all points in :
  • The accumulation points of are all points in :
  • The boundary points of are , , and :
  • has only one isolated point, namely .
  • The limit points of are all points in .

Example 4.1.9

Let .

  • .
  • .
  • has no isolated points.
  • Every point of is a limit point of .

Example 4.1.10

Let .

  • .
  • .
  • Every point of is an isolated point.
  • has no limit points.

Definition 4.2.1

Let be a set. We say that is (sequentially) compact if every sequence in has a subsequence that converges to a point .

In this course, we will frequently abbreviate “sequential compactness” to just “compactness”. However, there is a different concept which is called “compactness”, which is equivalent to sequential compactness for subsets of (and more generally for “metric spaces”), though the two aren’t equivalent in some more general settings. This is discussed further in Interesting Fact 4.2.7.

Example 4.2.2

Let be a finite set,

Then is compact. To see this, let be a sequence in . Then there must be some such that for infinitely many . Consequently, we can realize the constant sequence

as a subsequence .

Example 4.2.3

The set is not compact. To see this, take the sequence . This sequence diverges to , and it is not hard to see that any subsequence of it also diverges to . Hence it has no subsequence which converges.

Example 4.2.4

The set is not compact. To see this, take the sequence in . This sequence converges to , and hence so does every subsequence of it (Proposition 2.5.4). Hence, it has no subsequence converging to a point in .

Theorem 4.2.5 (Heine–Borel Theorem)

Let be a subset of . Then is compact if and only if is closed and bounded11.

Proposition 4.2.6 (Permanence properties of compact sets)

(i) For any finite collection of compact sets, , their union,

is compact.

(ii) For any arbitrary collection of compact sets , their intersection,

is compact.

Example 4.2.7

The correct definition of compactness is as follows: is compact if for any family of open sets in (indexed by any set ) such that

( is an “open cover” of ), there are such that

( is a “finite subcover”).

This definition makes sense in a much more general setting of topological spaces. Metric spaces are certain special topological spaces, which include all subsets of (and much more), and for these, sequential compactness is equivalent to compactness, a result which you will see in MAT 3120. The Heine—Borel Theorem, however, does not generalize to this setting. In the more general setting of topological spaces, compactness and sequential compactness are different notions, neither implying the other.

Continuous Functions

Definition 5.1.1

Let and let be a limit point of . Let and let . We write

if for every there exists such that, if and then .

The idea here is that when is close (but not equal) to , then should be close (and possibly equal) to . It makes sense that we restrict to being a limit point, because that is exactly the condition that tells us that there are points (arbitrarily) close to inside . Recall that a limit point may or may not be inside itself; however, if , then the value does not matter at all to the limit (or whether this limit exists).

Proposition 5.1.4

Let and let be a limit point. Let and let . If

then .

Proposition 5.1.5 (Sequential Characterization of Limits)

Let and let be a limit point. Let and let . Then if and only if for every sequence in which converges to , we have

Proposition 5.1.6 (Algebra of Limits)

Let and let be a limit point. Let be functions which all have limits at . Let . Then:

(i)

(ii)

(iii)

(iv) If for all and then

Theorem 5.1.7 (Squeeze Theorem)

Let and let be a limit point. Let be functions which satisfy

If

then

as well.

Definition 5.1.8

Let , let be a function and let . If is a limit point of , then we write

if for every there exists such that, if , , and

then

Likewise, if is a limit point of , then we write

if for every there exists such that, if , , and

then

Definition 5.2.1

Let and let be a point which is not isolated. Let . We say is continuous at if

If is an isolated point, then we say that any function is continuous at .

Making use of the definition of limit, we find that is continuous at if for any there exists such that, if and then . This reformulation says that is continuous if nearby points get sent to nearby images so it works whether or not is an isolated point.

Proposition 5.2.5

Let , let and be functions, and let . Suppose that is continuous at and is continuous at . Then is continuous at .

Proposition 5.2.6

Let and let be a limit point. Let be functions which are all continuous at . Let . Then:

(i) is continuous at .

(ii) is continuous at .

(iii) is continuous at .

(iv) If for all then is continuous at .

Definition 5.3.1

Let and let be a function. We say that is continuous (on ) if is continuous at for every .

Theorem 5.3.2

Let be compact and let be a continuous function. Then its image, , is also compact.

Corollary 5.3.3 (Extreme Value Theorem)

Let be compact and nonempty, and let be a continuous function. Then there exists such that for all ,

In other words, the image of is bounded above and below, and it attains its bounds.

Remark 5.3.4

If is not compact, then it is possible that is not bounded, or that it is bounded but it does not attain its bounds. Here are some examples.

Define by . Then is not bounded above.

Define by . Then , which is bounded, but has no maximum.

Theorem 5.3.5 (Intermediate Value Theorem)

Let be a continuous function. Let be any value between and . Then there exists such that .

The intermediate value theorem is a very powerful notion, since it gives an easy check as to whether an equation has a solution. For example, for any and , the function defined by is continuous. We have and, since as , we can find such that . Then the Intermediate Value Theorem immediately tells us that there exists such that , i.e., . In this way we can define roots.

Proposition 5.3.6

Let be a continuous function. Then for some .

Footnotes

  1. From the course MAT2125, elementary real analysis. ↩

  2. The argument is more subtle than simply picking two complex numbers. The real issue is that no ordering on can satisfy (O3) and (O4) simultaneously. Consider : either or . If , then by (O4), . But then as well, and by (O3), , i.e. , contradicting (O2). The case gives the same contradiction via . ↩

  3. This is typically taken as the axiom that distinguishes from . Depending on how you construct (Dedekind cuts, Cauchy sequences of rationals, etc.), completeness is either assumed or proved from the construction. ↩

  4. This follows from completeness. An equivalent way to state this is that there is no positive real number smaller than every , i.e. has no infinitesimals. It is however worth noting that is also Archimedean but not complete, so completeness is strictly stronger. ↩

  5. is dense in . See Dense order or Dense set. ↩

  6. In Prof. Tikuisis’s class, the concept of (weakly) increasing and decreasing sequences is the only important detail, and strictness is secondary. One notable, perhaps powerful, result about monotone sequences: they converge exactly when they are bounded. ↩

  7. It fails in : a bounded sequence of rationals can converge to an irrational, so you lose the guarantee of a convergent subsequence staying in your space. ↩

  8. In fact we only defined addition for two numbers at a time, but because of associativity we can always group two and two at each given time, so iterating, we can sum finitely many numbers. ↩

  9. If a series converges conditionally (but not absolutely), then for any target — or — there exists a rearrangement that converges to . Interesting theorem. ↩

  10. A common early misconception is that not open means closed. In fact, and are both open and closed (sometimes called “clopen”), while most sets are neither. ↩

  11. “Closed and bounded” as a characterization of compactness is specific to . In general metric spaces, closed and bounded does not imply compact. The standard counterexample is the closed unit ball in an infinite-dimensional Banach space. It’s closed and bounded but not (sequentially) compact. See also (Tao): Locally compact topological vector spaces (Terence Tao) ↩