The theorem is:
Suppose $M$ is an $R$-module, $T$ is an index set, and for each $t \in T$, $N_t$ is a submodule of $M$. If $\{N_t\}$ is a monotonic collection, $\cup_{t\in T} N_t$ is a submodule.
Is the following proof correct?
If $a, b \in \cup_{t\in T} N_t$, then $a, b \in N_t$ for some $t$. Then $a+b\in N_t$.
If $c\in \cup_{t\in T}N_t$, then $c\in N_t$ for some $t$, then $cr\in N_t$, then $cr\in \cup_{t\in T}N_t$.
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Similar proof for the other criteria to be a module.
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Is it somewhat too trivial to be a theorem?
Getting this out of unanswered-limbo
Yes, the proof is fine for proving that the union of a nondecreasing chain of submodules is a submodule. Although easy to prove, it turns out to be very useful. For example, it pops up when using Zorn's lemma to show rings with identity have maximal right ideals.