@Matthew You can find a few posts about this on the main, for example: $X_n\leq Y_n$ implies $\liminf X_n \leq \liminf Y_n$ and $\limsup X_n \leq \limsup Y_n$

It all depends on which definition of limit superior/inferior you're using - however, it can be shown that the various definitions are equivalent.

It should be straightforward to see that $\sup_{k\ge n} x_n \le \sup_{k\ge n} y_n$.

And then the inequality is true also for limit: $\lim_{n\to\infty}\sup_{k\ge n} x_n \le \lim_{n\to\infty}\sup_{k\ge n} y_n$.

@Matthew Do you mean "How do you enter MathJax in search?"

You can find some advice on searching here: How to search on this site?

To find the above question, I checked the posts tagged limsup-and-liminf+inequality and then chose the frequent tab.

@MartinSleziak I believe so. I want to know how to use MathJax to be able to enter the appropriate mathematical symbols etc when I am searching for something.

Probably the best way to search for MathJax expression is to use Approach0.

For example you can search for $\limsup x_n \leq \limsup y_n$ or for $\limsup_{n\to\infty} x_n \leq \limsup_{n\to\infty} y_n$.

Of course, you could use the same in the built-in search, but in that way you only find posts which have exactly the same expression. (In Approach0, searching for inequality between $x_n$ and $y_n$ returns also results where the sequences are denoted by $a_n$ and $b_n$.)

BTW there is also chatroom devoted specifically to searching.

Anyway, I think that most of the stuff I mentioned here is also explained in the FAQ post I linked above.

Since you have mentioned that you cannot upload images to chat - this is possible from 100 reputation points.