Let $E$ be a t.v.s. and $A, B \subseteq E$ with $A$ compact and $B$ closed. Then $A+B$ is closed
Solution 1:
To me, it is totally fine and it is a nice generalization to spaces that are not sequential spaces (for which the topology is not fully described by sequences). The proof is essentially the same (with some remarks like the use of the axiom of choice). As another reference I post here Duchamp Gérard H. E.’s answer that is very similar to your (correct) work: https://math.stackexchange.com/a/2079363/865323