If $\{x_{n}\}$ is a convergent sequence and $\{y_{n}\}$ is a bounded sequence then what can we say about $\{x_{n}+y_{n}\}$

Question

Suppose $\{x_{n}\}$ is a convergent sequence and $\{y_{n}\}$ is a bounded sequence then I was thinking about a series of questions.

1) Is $\{x_{n}+y_{n}\}$ convergent?, I thought of the following counterexample, Suppose $x_{n} = \frac{1}{n}$ and $y_{n} = (-1)^n$ then $x_{n}+y_{n} = (-1)^n + \frac{1}{n}$ is not convergent. Is this correct? also any simpler counter examples than this?

2) Is $x_{n}+y_{n}$ bounded? well if $\{x_{n}\}$ is convergent then $\{x_{n}\}$ is bounded then as $y_{n}$ is also bounded, so $\{x_{n}+y_{n}\}$ is also bounded.

3)Now does $\{x_{n}+y_{n}\}$ have a convergent subsequence, a bounded subsequence? how to show this?. I think of Cauchy sequence and a theorem related to this – "Every Cauchy sequence has a convergent subsequence" but it would not be necessary that $x_{n}+y_{n}$ be a Cauchy sequence and hence would have a Cauchy sequence. So $\{x_{n}+y_{n}\}$ need not have a convergent subsequence necessarily.

Answer 1

1. A simpler example would be $x_n=0$ and $y_n=(-1)^n$.
2. It is correct.
3. By the Bolzano-Weierstrass theorem, every bounded sequence has a convergent subsequence. And every subsequence of a bounded sequence is also bounded.