How to prove $\limsup (x_{n}+y_{n})=\lim x_{n}+\limsup y_{n}$?

Question

Given a convergent sequence $x_{n}$ and bounded sequence $y_{n}$ I need to prove that $\limsup (x_{n}+y_{n})=\lim x_{n}+\limsup y_{n}$, when $n$ tends to $\infty$.

I chose $z_{n}=x_{n}+y_{n}$, we know that $z_{n}$ is bounded as being sum of two bounded sequences, so from the Bolzano-Weierstrass Theorem, we know that there is a subsequence of $z_{n}$, let's call it $z_{n_{k}}$, that converges to $\limsup (x_{n}+y_{n})$. $y_{n_{k}}$ is bounded as well, so there is a convergent subsequence $y_{n_{k_{j}}}$. All this gives me that $\limsup (x_{n}+y_{n})\leq \lim x_{n}+\limsup y_{n}$,

What can I do for getting the other inequality?

Thank you and Good evening.

Answer 1

Simply take a convergent subsequence $y_{n_k}$ such that $\limsup y_n=\lim y_{n_k}$. Then $\limsup(x_n+y_n) \geq \lim (x_{n_k}+y_{n_k})= \lim x_n+\limsup y_n$

Answer 2

Hint:

Suppose $\{y_{n_k}\}$ is a convergent sub-sequence of $\{y_n\}$. What is the limit of $\{z_{n_k}\}$?

Answer 3

I moved this answer from another question as I was advised.

I decided to give here a version of the proof for the sentence from the comment using subsequences, since this technique seems more transparent and is not used on brought links above - I hope it will be useful to someone.

So, let's prove $$\exists\lim\limits_{n\to\infty}x_n\ \&\ \forall(y_n)_{n\in \mathbb{N}} \Rightarrow \varlimsup\limits_{n\to\infty}(x_n + y_n ) = \lim\limits_{n\to\infty}x_n + \varlimsup\limits_{n\to\infty}y_n$$ So, we have one limit point for $(x_n)_{n\in \mathbb{N}}$, let's denote it $x=\lim\limits_{n\to\infty}x_n$ and let $\{y^*\}$ be the set of limit points for $(y_n)_{n\in \mathbb{N}}$ i.e. for each $y^*$ exists $(y_{n_k})_{k\in \mathbb{N}}$ converged to it. Sequence $x_n + y_n$, therefore, have limit points $\{y^*+x\}$. For $\varlimsup\limits_{n\to\infty}y_n$ we have some $(y_{n_k})_{k\in \mathbb{N}}$ converged to it, so, $x_{n_k}+y_{n_k}$ converge to $\lim\limits_{n\to\infty}x_n + \varlimsup\limits_{n\to\infty}y_n$. On other hand this subsequence gives $\varlimsup\limits_{n\to\infty}(x_n + y_n )$.