So sum law of limits tell us
$\lim_{n\to\infty} (a_n+b_n)=X + Y$ if $\lim_{n\to\infty} a_n = X$ and $\lim_{n\to\infty} b_n = Y$
Here is my attempt to prove it.
Proof
Let $\frac{\epsilon}{2}>0$, then $\exists N_a,N_b:$
(1) $|a_n - X| < \frac{\epsilon}{2}$ whenever $n> N_a$
(2) $|b_n - Y| < \frac{\epsilon}{2}$ whenever $n> N_b$
Add (1) and (2) to get $|a_n - X| + |b_n-Y| < \epsilon \implies |a_n + b_n - (X + Y)| \leq |a_n - X| + |b_n-Y| < \epsilon \iff |a_n + b_n - (X + Y)| < \epsilon$
By setting $N=max\left \{ N_a,N_b \right \}$, we get the result.
I consulted an analysis book and they kinda did the same thing, but I think I worked backwards by adding (1) and (2).