$ \limsup_n (x_n+y_n) = \limsup_n x_n +\limsup y_n $

Question

I was wondering when is the following statement can be true?

Let $\{x_n\} ,\{y_n\} \subset \mathbb R $ are sequences. then $$ \limsup_n (x_n+y_n) = \limsup_n x_n +\limsup y_n $$

It is clear that it is true if the both of these sequences are convergent.

Also I can see that if $\{y_n \}=$ constant.then the statement above is true.

Also, Let $$ x_n = \left\{ \begin{array}{lr} 1 & \mbox{if } n \text{ is even } \\ 2 & \mbox{if } n \text{ is odd } \end{array} \right.$$ $$ y_n = \left\{ \begin{array}{lr} 2 & \mbox{if } n \text{ is even } \\ 1 & \mbox{if } n \text{ is odd } \end{array} \right.$$

Then, $$ \limsup_n (x_n+y_n) =3 \neq 4= \limsup_n x_n +\limsup y_n$$

So the statement could be false if the both of the sequences are nonconvergent. Also we will have a problem if one of them has $+ \infty $ as a limit and the other one has $- \infty $ as a limit so it isn't true for this case too.

So, only one case left to check, that is if we know that one of these sequence has a limit. And I couldn't find the answer for this case.

I was trying like this :

Since we know that $ \sup(A+B) =\sup A +\sup B $ and $\inf(A+B) =\inf A +\inf B $, then

$$ \limsup_n (x_n+y_n) = \inf_{n \in \mathbb N} \sup_{k \ge n} (x_n +y_n)= \inf_{n \in \mathbb N} \big( \sup_{k \ge n} (x_n) + \sup_{k \ge n} (y_n) \big)=\inf_{n \in \mathbb N} \sup_{k \ge n} (x_n) + \inf_{n \in \mathbb N} \sup_{k \ge n} (y_n) = \limsup_n x_n +\limsup y_n $$
so the statement is true. But here I didn't use the convergence of $\{x_n\}$ at all so it has to be true for any sequences. Since I have already given a counter example then I am sure that my proof is wrong, but I can't find the mistake in this proof. Also I still don't know if the statement is true if one of this sequence is convergent or not.

Any help?

Edit: From the comment I can see the mistake in my proof. So, I have one question now is this statement true when one of the sequences is convergent?

Answer 1

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 $\varlimsup\limits_{n\to\infty}(x_n + y_n )$.