Let $(s_n)$ and $(t_n)$ be sequences defined on $\mathbb{R}$.
Prove that lim sup $s_n$ + lim sup $t_n \geq$ lim sup $(s_n+t_n)$
Proof (can someone please verify it?): Set $\alpha = $ lim sup $s_n$ and $\beta = $ lim sup $t_n$
Let $\epsilon > 0$. Now, $\exists N_1$ such that $\forall n > N_1$, $s_n < \alpha + \frac{\epsilon}{2}$. Also, $\exists N_2$ such that $\forall n > N_1$, $t_n < \beta + \frac{\epsilon}{2}$
Set $N = $ max$\{N_1, N_2\}$. Then, $\forall n > N$, $s_n+t_n < \alpha + \beta + \epsilon$. So,
sup $\{s_n+t_n|n>N\} \leq \alpha + \beta + \epsilon$.
Since we can find such an $N$ for each $\epsilon > 0$, we conclude that lim sup $(s_n+t_n) \leq \alpha + \beta$