A question about double limits For my project, I’m trying to prove the following claim:
For all $\epsilon > 0$, $\exists X,Y \in S$ such that $\frac{F(X) + F(Y)}{F(X\cdot Y)} < 1 + \epsilon$.
To keep things short, I might skip over things that I don’t think need to be explained to get the gist of my question. $X= (x_1,\dots,x_n)$ and $Y=(y_1,\dots,y_m)$ are tuples that are in a set $S$ and $F(X)$ is a function that maps the tuple to a positive real value. $X \cdot Y$ denotes the concatenation of $X$ and $Y$, and $X^n$ denotes the concatenation of $X$, n times. Fix $X$ and let $(Y) = (Y_1,Y_2,\dots,Y_r,\dots)$ denote a sequence of tuples.
I have proven that
$\lim_{r\to \infty}{\lim_{n\to \infty}{\frac{F(X^n)+F(Y_r^n)}{F(X^n \cdot Y_r^n)}}} = 1$ for any fixed $X$.
My question is whether proving this double limit is enough to prove the existence of integers $r$ and $n$ such that  $\frac{F(X^n)+F(Y_r^n)}{F(X^n \cdot Y_r^n)} < 1 + \epsilon$ for any fixed $X$, or whether I need explicitly show a pair $X,Y$, which is produced for a given $\epsilon$.
 A: Yes, your limit statement is sufficient to prove the original claim.
Take any $\epsilon > 0$ and any sequence $X_0 \in S$. (I assume you know $S$ is not empty.) Applying your lemma,
$$ \lim_{r \to \infty} \lim_{n \to \infty} \frac{F(X_0^n)+F(Y_r^n)}{F(X_0^n \cdot Y_r^n)} = 1 $$
So there is a natural number $R$ so that for every $r \geq R$,
$$ \lim_{n \to \infty} \frac{F(X_0^n) + F(Y_r^n)}{F(X_0^n \cdot Y_r^n)} < 1 + \frac{\epsilon}{2} $$
In particular, it holds for $r=R$:
$$ \lim_{n \to \infty} \frac{F(X_0^n) + F(Y_R^n)}{F(X_0^n \cdot Y_R^n)} < 1 + \frac{\epsilon}{2} $$
And now this implies there is a natural number $N$ so that for every $n \geq N$,
$$ \lim_{n \to \infty} \frac{F(X_0^n) + F(Y_R^n)}{F(X_0^n \cdot Y_R^n)} < 1 + \frac{\epsilon}{2} + \frac{\epsilon}{2} $$
So it's true for $n=N$:
$$ \frac{F(X_0^N) + F(Y_R^N)}{F(X_0^N \cdot Y_R^N)} < 1 + \epsilon $$
So the claim is true for the particular pair $X = X_0^N$, $Y = Y_R^N$. The exact numbers $R$ and $N$ which make this valid could probably be found by examining the steps above, but if the limit statement is correct, we know they exist.
