TITLE: Another limit points task.
QUESTION [5 upvotes]: Let $x_n, y_n$ -- two numerical sequences with pairwise distinct terms (i.e $x_i \neq x_j$ and $y_i \neq y_j$ for all $i \neq j$).
We decided that $\lim_{n \to \infty} (x_ny_n) = \lim_{n \to \infty} (x_n + y_n) = 5$.
The question is to find all limit points of the set $A$, where:
$A = \{e^{x_n} + e^{y_n} \ | \ n \in \mathbb{N}\}$.
Definition of limit point which I know: $a \in X, A \subset X$ is limit point if for every $\varepsilon > 0$ punctured ball $B_{\varepsilon}(a)$ contains an element from $A$.
In our case $X = \mathbb{R}$.
I found that:
$x = \lim_{n \to \infty} (x_n) = \frac{5 + \sqrt{5}}{2}$
$y =\lim_{n \to \infty} (y_n) = \frac{5 - \sqrt{5}}{2}$
With solving a system
$xy=5$
$x+y=5$.
(We've got case with swapped values of x and y, but it's easy to show that we can do the same for them if we find solution).
My idea is to find $z = \lim z_n$, where $z_n = e^{x_n} + e^{y_n}$. And then we can say that limit point is only $z$ (from the limit defenition).
But I've got some troubles with finding this limit.

REPLY [3 votes]: $x_n$ and $y_n$ do not necessarily converge. For example let $u_n$ be strictly decreasing to $x$ and let $v_n$ be strictly increasing to $y;\,$ let $(x_n,y_n)=(u_n,v_n)$ when $n$ is even;$\,$ let $(x_n,y_n)=(v_n,u_n)$ when $n$ is odd.
Let $S_n=x_n+y_n$ and $P_n=x_ny_n.$ Let $D_n= S_n^2-4P_n.$ Then  $\{x_n,y_n\}=\{(S_n+\sqrt {D_n})/2,\,(S_n-\sqrt {D_n})/2\}.$ Let $b_n=\max(x_n,y_n)$ and $c_n=\min(x_n,y_n).$ Then $b_n\to x$ and $c_n\to y.$ Therefore $$e^{x_n}+e^{y_n}=e^{b_n} +e^{c_n}\to e^x+e^y=^{def} z.$$
So $z$ is the unique limit-point of $A$ if $A$ is an infinite set, which it could be.
It is possible that $A$ is a finite set. For example if $x<x_n<\ln z$ for every $n,$ with $x_n\to x,$ and $y_n=\ln  (z-e^{x_n}).$ Then $A=\{z\}.$