# Question about sums of limit superiors

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 $$(s_n+t_n) \leq \alpha + \beta$$

• Yes, it's all fine Commented May 17, 2014 at 4:38
• You made a small typo on $(t_n)$, but yes that should be fine. Commented May 17, 2014 at 5:11

Also, $\exists N_2$ such that $\boldsymbol{\forall n > N_1}$, $t_n < \beta + \frac{\epsilon}{2}$
This should read $\forall n > N_2$.