Two Proofs for Open Sets and Metric Subspaces

Question

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.

Answer 1

Both proofs are correct. The second proofs looks more complicated because it strives to prove openness of sets using the definition (i.e. a union of open sets is open). By appealing to the basis formulation of topology, the first proof saved a lot of effort.