Lim inf $ (x_n + y_n)$ less than or equal to lim inf $x_n +$ lim sup $y_n$ 
I have the question: Show that for bounded sequences   $(x_n)$ and   $(y_n)$: $$\liminf\limits_{n \rightarrow \infty} (x_n + y_n) \leq \limsup\limits_{n \rightarrow \infty} x_n + \liminf\limits_{n \rightarrow \infty} y_n$$

So far I have that:
$$\limsup\limits_{n \rightarrow \infty} (x_n + y_n) \leq \limsup\limits_{n \rightarrow \infty} x_n + \limsup\limits_{n \rightarrow \infty} y_n$$
and that
$$ \liminf\limits_{n \rightarrow \infty} (x_n + y_n) \ge \liminf\limits_{n \rightarrow \infty} x_n + \liminf\limits_{n \rightarrow \infty} y_n\tag{1}$$
But now I'm a bit stuck - I've tried using
$$ \limsup\limits_{n \rightarrow \infty} (-x_n) = - \liminf\limits_{n \rightarrow \infty} (x_n) \tag{2}$$
but I'm just going in circles.
There are similar questions to this but none that I have found address this specficially, any help?
 A: Using what you already have (superadditivity of $\liminf$ and relation between $\liminf$ and $\limsup$), you can justify the steps in the following relation which gives you the result $$\liminf_{n\to\infty}(x_n+y_n)−\limsup_{n\to\infty} x_n\overset{(2)}=\liminf_{n\to\infty}(x_n+y_n)+\liminf_{n\to\infty}(−x_n)\overset{(1)}\le \liminf_{n\to\infty} y_n$$
A: Not like that. Note that
$$
x_n + y_n \leq \left(\limsup\limits_{n \rightarrow \infty} x_n \right)+  y_n
$$
and apply $\limsup$ to both sides to conclude what you want.
A: (1) If $\lim\ t_n=t$, then $$ \lim\ \inf\ (x_n+t_n) = \lim\ \inf\
x_n + t
$$
Proof : Let $$\lim\ \inf\ (x_n+t_n)=\lim_m\ (x_{k_m} + t_{k_m}) $$
for some subsequence Here $$ \lim_m\ (x_{k_m} + t_{k_m}) =\lim_m \
x_{k_m} + t \geq \lim\ \inf\ x_n + t $$
And if $\lim_m \ x_{l_m}= \lim\ \inf\ x_n$, then $$ \lim\ \inf\ x_n+
t =\lim\ (x_{l_m} + t_{l_m}) \geq \lim\ \inf\ (x_n + t_n)
$$
(2) If $\lim\ \inf \ y_n=\lim_m\ y_{k_m}$, then $$ \lim\ \inf\
(x_n+y_n) \leq \lim\ \inf_m \ (x_{k_m} + y_{k_m}) $$ $$=\lim\ \inf\
x_{k_m} + \lim_m\ y_{k_m} =\lim\ \inf\ x_{k_m} + \lim\ \inf\ y_n
$$ $$\leq \lim\ \sup\ x_n  + \lim\ \inf\ y_n
$$
