# Prove $\lim \inf(X_n + Y_n) = \lim_{n→+∞} \ X_n +\ \lim \inf \ Y_n$

Assume $\lim_{n \rightarrow \infty} \ X_n$ exists. Prove that for any sequence $(Y_n)$, we have $$\lim \inf(X_n + Y_n) = \lim_{n→+∞} \ X_n + \lim \inf \ Y_n$$

How to start this problem?

• start by chewing the definitions, correcting $n\to1$ to $n\to\infty$ in your post, and perhaps letting $\lim X_n =L$, $\lim\inf Y_n = K$. Use that every subsequence of the $X_n$ has the same limit $L$. – Mirko Nov 26 '15 at 14:01
• Expanding on Mirko's comment: Let $Y_{n_i}$ be a convergent subsequence of $Y_n$. What can you say about the sequence $\{X_{n_i}+ Y_{n_i}\}$. What can you then say about $\liminf(X_n + Y_n)$? Then note that $Y_n = (-X_n) + (X_n + Y_n)$. – Paul Sinclair Nov 26 '15 at 16:11