4:05 AM
@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 $\limsup X_n \leq \limsup Y_n$
4:27 AM
To find the above question, I checked the posts tagged limsup-and-liminf+inequality and then chose the frequent tab.
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 $b_n$.)
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.
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