Here are some practice problems you might use in preparing for the final exam. I’m sorry to say that I won’t have a chance to post solutions to these problems before the final. I will try to post solutions to the final itself by the evening of May 26, but I can’t make that a definite promise.

1. Let E = (0, 1), a subset of ℝ with the usual metric. Prove directly from the definition (without using the Heine-Borel theorem) that E is not compact. (This means you need to find a collection {U_{α}} of open subsets of ℝ so that E is contained in the union of all the {U_{α}}, but E is not contained in the union of any finite subcollection of the {U_{α}}.)

2. Let X be the metric space of all rational numbers in [0, 1]. Find a subset E of X such that E ≠ X, E ≠ ∅, but E is both open and closed in X.

3. Give an example of a subset of ℝ having exactly two limit points.

4. Give an example of a real-valued differentiable function f on ℝ for which f′ is not continuous.

5. (20 points) X is a metric space and

f: X → X

is a function from X to X. A fixed point of f is a point x ∈ X such that f(x) = x. Prove that every continuous function from [0, 1] to [0, 1] has a fixed point. (Hint: you want to show that the continuous function f(x) − x is equal to zero somewhere. Use the Intermediate Value Theorem.)

6. Suppose that f: X → X is any continuous function, and that x_{0} ∈ X. Define a sequence x_{1}, x_{2}, x_{3} … of points in X by

x_{n+1} = f(x_{n}) (n ≥ 0).

Prove that if the sequence {x_{n}} converges to a limit point x ∈ X, then f(x) = x.

7. This problem concerns Riemann sums for integrating the function x on the interval [a, b]. You may need to use the formula

∑_{j=1}^{n} j = n(n + 1)/2.

For each positive integer n, consider the partition of [a, b] into n equal parts:

P_{n} = (a = x_{0} < x_{1} < ⋯ < x_{n} = b), x_{i} = a + i(b − a)/n.

a) Calculate the upper sum U(P_{n}, x).

b) Calculate the lower sum L(P_{n}, x).

c) Deduce from these two calculations (not using the Fundamental Theorem of Calculus) that ∫_{a}^{b} xdx = (b^{2} − a^{2})/2.

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8. Suppose that f is a continuous function on [0, 2π]. Recall that the mth Fourier coefficient of f is by definition

c_{m}(f) = \frac{1}{2π} ∫_{0}^{2π} f(x)e^{−imx}dx (m ∈ ℤ).

Suppose that the series of real numbers

∑_{n=1}^{∞} |c_{n}| + |c_{−n}|

converges.

a) Deduce that the Fourier series ∑_{m=−∞}^{∞} c_{m}e^{imx} converges uniformly to a continuous function F (x).

b) Show that c_{m}(F) = c_{m}(f).

9. Suppose {s_{n}} is a sequence of real numbers. Define a new sequence {σ_{n}} by

σ_{n} = \frac{1}{n}(s_{1} + ⋯ s_{n}),

the average of the first n elements of the first sequence.

a) Prove that if lim_{n→∞} s_{n} = s, then lim_{n→∞} σ_{n} = s.

b) Find an example of a sequence {s_{n}} that has no limit, for which {σ_{n}} converges. (Hint: make {s_{n}} bounce back and forth between two values.)

c) Can there be an unbounded seqence {s_{n}} for which {σ_{n}} converges?