Exercises from Real Analysis with worked solutions.

Exercise 8. Question 4.2.A
§4.2HW 2
(a) If (xn)n=1∞ is a sequence in Rn with lim⁡n→∞xn=a,show that lim⁡n→∞∥xn∥=∥a∥.(b) Show by example that the converse is false.\text{(a) If $(x_n)^{\infty}_{n=1}$ is a sequence in $\mathbb{R}^n$ with $\lim_{n \to \infty} \mathbf{x_n} = \mathbf{a}$,} \\ \text{show that $\lim_{n \to \infty} \lVert \mathbf{x}_n \rVert = \lVert \mathbf{a} \rVert$.} \\[0.5em] \text{(b) Show by example that the converse is false.}
Exercise 9. Question 4.2.B
§4.2HW 2
Show that if (xk) is a sequence in Rn such that∑k≥1∥xk−xk+1∥<∞,then (xk) is a Cauchy sequence.\text{Show that if $(\mathbf{x}_k)$ is a sequence in $\mathbb{R}^n$ such that} \\ \text{$\sum_{k \geq 1} \lVert \mathbf{x}_k - \mathbf{x}_{k+1} \rVert < \infty$,} \\ \text{then $(\mathbf{x}_k)$ is a Cauchy sequence.}
Exercise 10. Question 4.2.D
§4.2HW 2
Let x0∈Rn and R>0. Prove that{x∈Rn:∥x−x0∥≤R} is complete.\text{Let $x_0 \in \mathbb{R}^n$ and $R > 0$. Prove that} \\ \text{$\{x \in \mathbb{R}^n : \|\mathbf{x} - \mathbf{x_0}\| \leq R \}$ is complete.}
Exercise 11. Question 4.2.E
§4.2HW 2
Let M be a subspace of Rn.(a) Let {v1,…,vm} be an orthonormal basis for M.Formulate an analogue of Lemma 4.2.3 for M andprove it.(b) Prove that M is complete.\text{Let $M$ be a subspace of $\mathbb{R}^n$.} \\[0.5em] \text{(a) Let $\{\mathbf{v}_1, \dots, \mathbf{v}_m\}$ be an orthonormal basis for $M$.} \\ \text{Formulate an analogue of Lemma 4.2.3 for $M$ and} \\ \text{prove it.} \\[0.5em] \text{(b) Prove that $M$ is complete.}
Exercise 12. Question 4.2.F
§4.2HW 2
Let v0=(x0,y0) with 0<x0<y0. Definevn+1=(xn+1,yn+1)=(xnyn, xn+yn2)for all n≥0.(a) Show by induction that0<xn<xn+1<yn+1<yn.(b) Then estimate yn+1−xn+1 in terms of yn−xn.(c) Thereby show that there is a number c such thatlim⁡n→∞vn=(c,c).\text{Let $\mathbf{v}_0 = (x_0, y_0)$ with $0 < x_0 < y_0$. Define} \\ \mathbf{v}_{n+1} = (x_{n+1}, y_{n+1}) = \left( \sqrt{x_n y_n}, \ \frac{x_n + y_n}{2} \right) \\ \text{for all $n \geq 0$.} \\[0.5em] \text{(a) Show by induction that} \\ \text{$0 < x_n < x_{n+1} < y_{n+1} < y_n$.} \\ \text{(b) Then estimate $y_{n+1} - x_{n+1}$ in terms of $y_n - x_n$.} \\ \text{(c) Thereby show that there is a number $c$ such that} \\ \text{$\lim_{n \to \infty} \mathbf{v}_n = (c,c)$.}
Exercise 13. Question 4.2.H
§4.2HW 2
Let T=[5/4−1/43/41/4]. Set xn=Tn(1,0) for n≥1.(a) Prove that (xn) converges and find the limit y.(b) Find an explicit N so that ∥xn−y∥<1210−100 forall n≥N.\text{Let } T = \begin{bmatrix} 5/4 & -1/4 \\ 3/4 & 1/4 \end{bmatrix}\text{. Set $\mathbf{x}_n = T^n(1,0)$ for $n \geq 1$.} \\[0.5em] \text{(a) Prove that $(\mathbf{x}_n)$ converges and find the limit $\mathbf{y}$.} \\ \text{(b) Find an explicit $N$ so that $\|\mathbf{x}_n - \mathbf{y}\| < \frac{1}{2} 10^{-100}$ for} \\ \text{all $n \geq N$.}
Exercise 14. Additional Problem
HW 2
If (xk)⊆Rn converges to a∈Rn and ∥xk∥≤81 for allk∈N,then ∥a∥≤81.\text{If $(\mathbf{x}_k) \subseteq \mathbb{R}^n$ converges to $\mathbf{a} \in \mathbb{R}^n$ and $\|\mathbf{x}_k\| \leq 81$ for all} \\ \text{$k \in \mathbb{N}$,} \\ \text{then $\|\mathbf{a}\| \leq 81$.}
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