Assume that $\{y_n\}$ converges to $l$. Prove that $\liminf{x_n + y_n} = \liminf{x_n} + l$

Question

Let $\{x_n\}$ and $\{y_n\}$ be two bounded sequences. Assume that $\{y_n\}$ converges to $l$. Prove that $\liminf{(x_n + y_n)} = \liminf{x_n} + l$.

In the exercise we were asked to show that $\liminf{(x_n + y_n)} \geq \liminf{x_n} +\liminf{y_n} $.

$\{y_n\}$ converges to $l$ implies $\liminf{y_n}=l$ so that we have: $$\liminf{(x_n + y_n)} \geq \liminf{x_n} + l$$ How do I get the reverse inequality?

Answer 1

Use the same inequality again. This time replace $x_n$ with $x_n+y_n$ and replace $y_n$ with $-y_n$ (which has limit $-l$)