Lim inf $ (x_n + y_n)$ less than or equal to lim inf $x_n +$ lim sup $y_n$

Question

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?

Answer 1

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 $$\liminf_{n\to\infty}(x_n+y_n)−\limsup_{n\to\infty} x_n\overset{(2)}=\liminf_{n\to\infty}(x_n+y_n)+\liminf_{n\to\infty}(−x_n)\overset{(1)}\le \liminf_{n\to\infty} y_n$$

Answer 2

Given $\varepsilon>0$, we have $$ x_n + y_n \le\limsup\limits_{m \rightarrow \infty} x_m+ y_n+\varepsilon $$ for any large $n$. Applying $\liminf$ to both sides and letting $\varepsilon\to0$ gives the desired inequality.

Answer 3

(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 $$ \lim\ \inf\ (x_n+y_n) \leq \lim\ \inf_m \ (x_{k_m} + y_{k_m}) $$ $$=\lim\ \inf\ x_{k_m} + \lim_m\ y_{k_m} =\lim\ \inf\ x_{k_m} + \lim\ \inf\ y_n $$ $$\leq \lim\ \sup\ x_n + \lim\ \inf\ y_n $$