# Problem Set 3

1. Easy
Let $$\left\{x_{k}\right\}$$ and $$\left\{y_{k}\right\}$$ be sequences such that $$x_{k}\to a$$ and $$y_{k} \to b$$. Show that $$x_{k}+y_{k}\to a+ b$$ and $$x_{k}y_{k}\to ab$$.

2. Easy
Let $$x_{k} = \frac{3k+4}{k-5}$$. Given $$\epsilon > 0$$, find an integer $$K$$ such that $$\vert x_{k}-3\vert < \epsilon$$ for all $$k > K$$.

3. Easy
Let $$\left\{x_{k}\right\}_{k=1}^{\infty}\subseteq\mathbb{R}^{n}$$ be a sequence, and let the components be given by $$x_k = (x_k^1, x_k^2, \ldots, x_k^n)$$. Prove that $$x_k \to x = (x^1,x^2,\ldots,x^n)$$ if and only if $$x_{k}^{i}\to x^{i}$$ for all $$i$$. In other words, a sequence in $$\mathbb{R}^{n}$$ converges if and only if each of its components converges.

4. Medium
Find an example of a sequence $$\left\{x_{k}\right\}$$ such that $$\vert x_{k+1} - x_{k} \vert \to 0$$, but $$\left\{x_{k}\right\}$$ is not Cauchy.

5. Medium
Let $$S\subseteq\mathbb{R}$$, and set $$L = \inf S$$. Show there exists a sequence $$\left\{x_k\right\}$$ converging to $$L$$.

6. Medium
Prove whether each of the following sequences converges or does not have a limit
1. $$((-1)^k, 0)$$
2. $$(\cos(\pi k/2), \sin( \pi k/2))$$
3. $$\left(\frac{\cos\pi k}{k},\frac{\sin \pi k}{k}\right)$$
4. $$\left( (1+1/k)\cos(\pi k/2), (1+1/k)\sin(\pi k/2)\right)$$
5. $$(\cos(Ck),\sin(Ck))$$ where $$C/\pi$$ is irrational [Hard]
6. $$(k^2, 0, 1/k)$$

7. Medium
If $$\left\{x_{k}\right\}\subseteq\mathbb{R}^{n}$$ is a Cauchy sequence and there exists a subsequence $$\left\{x_{k_{n}}\right\}$$ such that $$x_{k_{n}}\rightarrow x$$, then $$x_{k} \to x$$.

8. Hard
Construct a sequence $$\left\{z_{k}\right\}_{k=1}^{\infty}\subseteq \mathbb{R}^{2}$$ with the property that for any point $$z = (x,y)$$ such that $$x^2+y^2 = 1$$, there exists a subsequence $$\left\{z_{k_{n}}\right\}_{n=1}^{\infty}$$ such that $$z_{k_{n}}\to z$$.

9. Hard
Construct a sequence $$\left\{x_{k}\right\} \subseteq \mathbb{R}^{2}$$ with the property that for any $$x \in \mathbb{R}^{2}$$, there exists a subsequence $$\left\{x_{k_{n}}\right\}$$ which converges to $$x$$.

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