2014 Spring Comprehensive in Analysis

Problem 1.

Suppose that is differentiable on and has distinct zeros in . Prove that has at least zeros in .

Proof.

Let the distinct zeros of be

where each and .

For each , the function is continuous on and differentiable on . Since

Rolle's theorem gives a point such that

The intervals are disjoint, so the points are distinct. Hence has at least zeros in .

Problem 2.

Let be a sequence of points in such that

Prove that is a convergent sequence in .

Proof.

The condition implies that the numerical series

converges. Therefore its tails tend to . That is, for every , there exists such that whenever ,

For such and , the triangle inequality gives

Thus is a Cauchy sequence in . Since is complete, converges in .

Problem 3.

Show that the sequence defined recursively by

converges, and find its limit.

Proof.

First we show that for all . Since , suppose . Then

Thus for all .

Next, for any , we have

because both sides are positive and

Applying this with gives

for all . Therefore is decreasing and bounded below by . Hence it converges. Let

Since , we have . Passing to the limit in

we obtain

Squaring gives

Therefore

Hence converges to .

Problem 4.

(a) State Stokes' Theorem.

(b) Evaluate the following integral:

where

Proof.

(a) Stokes' Theorem says the following. If is an oriented smooth manifold with boundary , and is a smooth differential form of degree one less than , then

where carries the induced boundary orientation.

(b) Let

Then

Also,

Since

we get

Finally,

Therefore

By Stokes' Theorem,

The region is the ellipsoid with semiaxes . Put

Then and

Hence

By symmetry,

Using spherical coordinates,

Thus

Therefore

So the value of the integral is

Problem 5.

Let be a subset of such that every point is an isolated point. Prove that is at most countable.

Proof.

Since every point is isolated, for each there exists such that

Let be the collection of all open balls in whose centers have rational coordinates and whose radii are positive rational numbers. This is a countable basis for the topology of .

For each , since is open and contains , there exists a basis element such that

Then

We claim the assignment is injective. Indeed, if and , then , so . Therefore cannot equal , because .

Thus injects into the countable set . Hence is at most countable.

Problem 6.

Let be the space of all real-valued continuous functions on with metric

Let

Prove that is a compact subset of .

Proof.

Define

We first show that is continuous with respect to the metric . For , by the mean value theorem applied to the sine function,

for every . Therefore

Thus is continuous.

Since is compact, the image is compact in . Also, because sine is -periodic,

Hence

and therefore is compact.

Problem 7.

Let be a Riemann integrable function on . Prove that

Proof.

We first prove the result for a step function. Suppose

up to endpoints. Then

For each interval,

which tends to as . Hence

Now let be Riemann integrable on . Given , by Riemann integrability there exists a step function such that

Then

Since ,

For the fixed step function , we already proved that

Therefore, for all sufficiently large ,

Hence, for all sufficiently large ,

Since was arbitrary,

Problem 8.

Let

Prove that is uniformly continuous on .

Proof.

The function is differentiable on , and

Therefore

For , we have

because

Thus

for all .

Now take any . Define

Then

Hence

Thus is Lipschitz continuous with Lipschitz constant . In particular, is uniformly continuous on .

Problem 9.

Let be a differentiable function on such that

for some positive constant . Prove that

for all .

Proof.

Assume without loss of generality that . By the fundamental theorem of calculus,

Therefore, by the Cauchy-Schwarz inequality,

Thus

Since , we obtain

The case is immediate. Hence the estimate holds for all .