HOMEWORK 4 — DUE OCT 17TH

MATH 25

There are three sections: A,B and C (plus a fourth “optional” one which won’t be graded). Please return the three parts separately to the proper CA.

A. Problems graded by Jennifer

A.1. A regularization operator. Let {x_{n}}_{n≥1} be a sequence of real numbers. Set

y_{n} = \frac{1}{n} (x_{1} + x_{2} + ⋯ + x_{n})

If y_{n} → y for some y ∈ R, we say that {x_{n}}_{n≥1} Cesàro-converges to y.

(1) Show that if x_{n} converges to x, then it also Cesàro-converges to x.

(2) Find the Cesàro-limit of the sequence {x_{n} = (−1)^{n}}.

A.2. Extending continuous functions. Let f : [0, 1) → R be a continuous function. Show that one can extend f to a continuous function on [0, 1] if and only if f is uniformly continuous on [0, 1).

A.3. Discontinuities of non-decreasing functions. Let f : R → R be a non-decreasing function.

(1) Show that f is continuous at x ∈ R if and only if sup_{y<x} f(y) = f(x) and inf_{y>x} f(y) = f(x) (draw a picture!).

(2) Prove that the set of x ∈ R at which f is not continuous is countable.

B. Problems graded by Benjamin

B.1. Writing a number in base b. Let b ≥ 2 be an integer, and choose n ∈ N.

(1) Show that one can write n = ∑_{k=0}^{ℓ} x_{k}b^{k} with x_{k} ∈ {0, 1, ⋯, b − 1} and some ℓ ≥ 0 in exactly one way.

(2) Choose x ∈ [0, 1) now. Show that one can write x = ∑_{k=1}^{+∞} x_{k}b^{−k} with x_{k} ∈ {0, 1, ⋯, b − 1} in exactly one way, if we impose the additional condition that for every N ≥ 0, there exists n ≥ N such that x_{n} ≠ b − 1.

2 MATH 25

B.2. The digits of 2^{n}. Let log_{2} stand for the logarithm in base 2: if x > 0, log_{2}(x) is the unique real number such that 2^{log_{2}(x)} = x.

(1) Prove that log_{2}(10) is irrational (hint: what can you say if 2^{p/q} = 10?).

(2) Show that there is a number n ≥ 1 such that when one writes 2^{n} in base 10, the first few digits are 121 219 76....

B.3. Lévy’s chords lemma. Let f : [0, 1] → R be a continuous function such that f(0) = f(1) and choose q ∈ Z_{≥1}.

Prove that there exists x ∈ [0, 1 − 1/q] such that f(x + 1/q) = f(x).

C. Problems graded by Inna

C.1. Lim sup and lim inf. Let {x_{n}}_{n≥1} be a bounded sequence of real numbers, and set y_{m} = sup_{n≥m} x_{n}.

(1) Prove that {y_{m}}_{m≥1} is a non-increasing sequence bounded from below, so that it converges. Define lim sup x_{n} to be the limit of y_{m};

(2) Give a similar definition of lim inf x_{n};

(3) Prove that lim sup x_{n} ≥ lim inf x_{n} and that {x_{n}}_{n≥1} has a limit if and only if lim sup x_{n} = lim inf x_{n};

(4) Recall that if A(x) is the set of accumulation values of {x_{n}}_{n≥1}, then A(x) is closed. Why does it have a largest element? Prove that ℓ = lim sup x_{n} is the largest element of A(x) and that for any ϵ > 0 the set {n ∈ Z_{≥1} x_{n} > ℓ + ϵ} is finite;

(5) Give the corresponding statements for lim inf.

C.2. A special class of sequences. Let {α_{n}}_{n≥1} be a sequence of real numbers bounded from below and such that (n + m)α_{n+m} ≤ nα_{n} + mα_{m}.

Show that there exists α ∈ R such that α_{n} → α. Hint: use the preceding exercise.

C.3. Rearranging a series. Let ∑_{n=1}^{∞} a_{n} be a convergent series, which converges to ℓ ∈ R. If ϕ : Z_{≥1} → Z_{≥1} is a bijection, we consider the rearranged series ∑_{n=1}^{∞} a_{ϕ(n)}.

(1) Show that if ∑_{n=1}^{∞} a_{n} is absolutely convergent, then any rearrangement ∑_{n=1}^{∞} a_{ϕ(n)} also converges to ℓ.

(2) Assume now that ∑_{n=1}^{∞} a_{n} is not absolutely convergent Let P and N be the subsets of Z_{≥1} such that m ∈ P if a_{m} ≥ 0 and m ∈ N if a_{m} < 0. Prove that P and N both contain infinitely many elements ≠ 0, and that furthermore ∑_{k∈P} a_{k} → +∞ and ∑_{m∈N} a_{m} → −∞.

HOMEWORK 4 — DUE OCT 17TH 3

(3) Let z be any real number. Set ϕ(1) = 1 and define ϕ(n) by recurrence in the following way: if ∑_{k=1}^{n−1} a_{ϕ(k)} > z then set ϕ(n) to be any element of N which has not already been chosen. Otherwise, set ϕ(n) to be any element of P which has not already been chosen.

(4) Prove that the rearranged series ∑_{n=1}^{∞} a_{ϕ(n)} converges to z.

D. Supplementary problems

D.1. Dilating maps. Let (E, d) be a compact metric space and let f : E → E be a continuous map such that d(f(x), f(y)) ≥ d(x, y) for all x, y ∈ E. Show that f is a bijection and that d(f(x), f(y)) = d(x, y) for all x, y ∈ E.

D.2. Steinitz’ theorem. Let {a_{n}}_{n≥1} and {b_{n}}_{n≥1} be two sequences of real numbers such that ∑_{n≥1}(a_{n}, b_{n}) = (0, 0). Let S be the set of all possible limits in R^{2} of ∑_{n≥1}(a_{ϕ(n)}, b_{ϕ(n)}) when ϕ runs among all bijections ϕ : Z_{≥1} → Z_{≥1}. Show that S is either {(0, 0)} or a line or R^{2} itself. Generalize to higher dimensions.