Topics in Analysis Examples Sheet 4 W. T. G.

1. (i) For each n let F_{n} be the nth Fibonacci number, with F_{1} = F_{2} = 1. Prove the identities F_{2n+1} = F_{n}^{2} + F_{n+1}^{2} and F_{2n} = F_{n}(F_{n−1} + F_{n+1}). Let x_{n} stand for the ratio F_{n+1}/F_{n}. Use the above identities to express x_{2n} in terms of x_{n}.

(ii) If we set y_{k} = x_{2^{k}}, then we have expressed each y_{k} as a simple rational function of y_{k−1}. Explain why the sequence (y_{k}) converges very rapidly to the golden ratio. How does this relate to another method that produces rapid convergence?

2. What are the eigenvalues of the matrix ( \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} )? What are the connections between this, Fibonacci numbers and continued fractions?

3. Assuming the continued-fraction expansion of tan(x) given in lectures, derive an expansion for tanh(x). Deduce that

\frac{e + 1}{e − 1} = 2 + \frac{1}{6 + \frac{1}{10+ \frac{1}{14+...}}}

4. Let X and Y be topological spaces, let x ∈ X and let f : X → Y. Say that f is continuous at x if for every neighbourhood N of f(x) there exists a neighbourhood M of x such that f(M) ⊂ N. Prove that f is continuous if and only if it is continuous at every x ∈ X.

5. Let X = {0} ∪ {n^{−1} : n ∈ ℕ} and let X have the subspace topology inherited from ℝ. Which subsets of X are open? Which are closed? Which are compact?

6. The open sets in the cofinite topology on an infinite set X are the empty set and all subsets Y of X such that X \ Y is finite. Write down and prove necessary and sufficient conditions for a function f : X → X to be continuous under this topology.

7. Let X be a Hausdorff topological space and let x_{1}, …, x_{n} be n distinct points in X. Prove that there exist pairwise disjoint open sets U_{1}, …, U_{n} with x_{i} ∈ U_{i} for every i.

8. Let X be a complete metric space and let Y be another metric space that is homeomorphic to X. Must Y be complete?

9. Let 𝕋 be the unit circle {z ∈ ℂ : |z| = 1} with the obvious topology coming from ℂ. Define an equivalence relation ∼ on ℝ by x ∼ y iff x − y is an integer. Prove that 𝕋 is homeomorphic to ℝ/ ∼ with the quotient topology.

10. Prove that every metric space is normal. Find an example of a Hausdorff topological space that is not normal.

11. Let X be a subset of ℝ. A point x in X is called isolated if there exists some δ > 0 such that |x − y| ⩾ δ for every other y ∈ X. X is called perfect if it is closed and has no isolated points. Prove that a non-empty perfect set must be uncountable. (Hint: consider X as a metric space in its own right, and don’t forget the Baire category theorem.)

12. Let f : ℝ → ℝ be a continuous function with the property that for every x > 0 we have f(nx) → 0 as n → ∞. Prove that f(x) → 0 as x → ∞.