If $x_n$ and $y_n$ are bounded, show that $\lim \inf (x_n + y_n) \leq \lim \inf x_n + \lim \sup y_n$

Question

Suppose you have $x_n$ and $y_n$ that are bounded. Show then that:

$$\lim \inf x_n + \lim \inf y_n \leq \lim \inf (x_n + y_n) \leq \lim \inf x_n + \lim \sup y_n $$

So to show this part: $\lim \inf x_n + \lim \inf y_n \leq \lim \inf (x_n + y_n)$. I defined three sets: $$X_n = \{ x_k | k \geq n \} $$ $$Y_n = \{y_k | k \geq n \} $$ and $$XY_n = \{x_k + y_k | k \geq n \} $$ Let $n\in \mathbb{N}$. We then have $$XY_n \subset X_n + Y_n$$, thus $$ \inf X_n + \inf Y_n \leq \inf XY_n$$ thus $$ \lim\inf (X_n + Y_n) \leq \lim \inf XY_n \iff \lim \inf X_n + \lim \inf Y_n \leq \lim \inf (X_n + Y_n)$$.

Now all is left is to show that $$ \lim \inf (x_n + y_n) \leq \lim \inf x_n + \lim \sup y_n .$$

Any ideas how to proceed?

Answer 1

Hint: We observe \begin{align*} \liminf(x_n) &= \liminf(x_n + y_n - y_n) \\ &\ge \liminf(x_n + y_n) + \liminf(-y_n) \\ &= \liminf(x_n + y_n) - \limsup (y_n). \end{align*}

Answer 2

HINT

Use the fact that:

$$\liminf_{n \to \infty}x_n=\sup_n\{\inf x_k:k \geq n\}$$