This is my first post here, looking forward to discussing math stuff with everyone. No one in my family knows anything about Real Analysis, so this will be the first time I ever actuallytalkto anyone about this. I've been plugging along, and I've proved every theorem up to this one, in chapter two section four on sequences of real numbers.

Royden defines the limit superior as follows:

$\displaystyle \inf_{n} \sup_{k \geq n} x_k\ $

He abbreviates the limit superior as:

$\displaystyle \overline{lim}\ x_n$

He defines the limitinferioras sup inf x, as you would expect. The theorem I am trying to prove is:

$\displaystyle \overline{lim}\ x_n\ + \underline{lim}\ y_n\ \leq\ \overline{lim}\ x_n\ +\ y_n$.

I have already proved the Cauchy Criterion and some other useful stuff about cluster points, so I feel like I have a pretty good grasp on this. If someone could just point me in the right direction, I'd appreciate it! I have worked out a working proof, which requires one lemma I don't know how to prove. Anyway, I feel like there's a simpler way to go about this that I'm missing.

By the way, I have proved the epsilon-N-k definitions of the limit superior, so feel free to use those in your explanation. Thanks again, in advance.

--Jake