I have two proofs for the following theorem:
Let $(S, d)$ be a metric subspace of $(M, d)$, and let $X$ be a subset of $S$. Then $X$ is open in $S$ if and only if $X = A \cap S$ for some set $A$ that is open in $M$.
The first proof:
The second proof:
Note that the second proof proves the same theorem but written differently:
Let $(Y, d)$ be a metric subspace of $(X, d)$. Then $E \subset Y$ is open in $(Y, d)$ if and only if there exists a set $G$ that is open in $(X, d)$ such that $E = G \cap Y$.
My question is whether these two proofs are equivalent or is one of them more correct than the other? The reason I ask is because of the considerable difference in length! If the first proof is enough, then I would definitely prefer it to the second proof.