Lemma
If $s_{n,k}$ are real numbers for which $n\in\mathbb{N}$, $k\in I$ with $I$ a generic set of indices, $s_{n,k}\leq s_{n+1,k}$ for all $k\in I$, then: $$\lim_{n\to\infty}\sup_{k\in I}s_{n,k}=\sup_{k\in I}\lim_{n\to\infty}s_{n,k}.$$
Proof
We want to compare $\lim_n\sup_ks_{n,k}$ and $\sup_k\lim_ns_{n,k}$. The sup of the limits is greater than any limit. Therefore it is greater than any $s_{n,k}$. Therefore it is greater than any sup, implying it is greater than the limit of the sups. So: $$(1)\qquad \sup\lim\geq\lim\sup.$$ The sequences $(s_{n,k})_{n\in\mathbb{N}}$ we are working with are monotonic nondecreasing in $n$, so $s_{n,k}\leq s_{n+1,k}$ for all $k,n$. Therefore the sequence of the sups is also monotone, so the limit of the sups is greater than all of the sups, so greater than all $s_{n,k}$, so greater than any $\lim_ns_{n,k}$, and so also of their sup. This proves the other inequality: $$(2)\qquad \lim\sup\geq\sup\lim.$$ By combining (1) and (2), we get the lemma.
By a similar argument, one can probably deduce the inf swaps with the limits in case of non-increasing monotonicity ($s_{n,k}\geq s_{n+1,k}$ for all $k\in I,n\in\mathbb{N}$).
I came up with these (the lemma with proof and the isolated sentence above) in my study of Measure Theory, and this is an extract of my answer to this question. Have these two lemmas ever been proved? I would expect so, since they are not that hard to prove, and a particular case was used by my teacher to prove the monotone convergence theorem for abstract integrals. Have they? Do they have a name? Also, are these proofs valid? And is there any other suitable tag for this question?
Edit: When I say I'd expect them to have been proved, I don't mean to say they are precious or anything. I'm merely saying, well, I'm not a super-mathematician, and I'm undergraduate, so if I proved them, some other more well-known person is bound to have observed this already. Just for clarity's sake.
