Let $L, M$ be real numbers, and let $s_n$ and $t_n$ be sequences such that $L \leq s_n \leq M$ and $L \leq t_n \leq M$ for all $n$ Define:

Question

I understand Part (a) just fine, but I am lost as to how to prove the two inequalities given. Can anyone help me out here?

Answer 1

Let $k$ be a positive integer. For all $n \ge k$,

$$s_n + t_n \ge \inf_{n\ge k} s_n + \inf_{n\ge k} t_n = L_{S,k} + L_{T,k}.$$

Thus $L_{S,k} + L_{T,k}$ is a lower bound for $U_k$. Consequently,

$$\inf_{n\ge k} (s_n + t_n) \ge L_{S,k} + L_{T,k},$$

that is,

$$L_{U,k} \ge L_{S,k} + L_{T,k}.$$

This proves $(b)$. To prove $(c)$, use the result in $(b)$ and take the limit as $k\to \infty$.