on Limits and sequences proofs of a closed subspace of the hilbert space.

Question

$M$ is a closed subspace of the Hilbert space $H$ and $ x\in H$

My book states these two claims.

(1) If $d = \inf_{y \in M} \|x - y \|^2 $then there is a sequence of elements $\{y_n\}$ of $M$ such that $\| y_n - x \|^2 \rightarrow d $

(2) $(y_m + y_n)/2 \in M$.

Now by def $M$ contains all it's limit points ,i.e., if $x_n \in M$ and $\| x_n - x \| \rightarrow 0 \implies x \in M$. But knowing this I can't manage to prove why the two claims follow.

Answer 1

(1) By the definition of infimum, $d+\frac1n$ is not a lower bound for $\{\|x-y\|^2 : y\in M\}$, that is, there exists $y_n\in M$ such that $\|x-y_n\|^2<d+\frac1n$. Then $d\le \|x-y_n\|^2 < d+\frac1n$; apply the squeeze theorem.

(2) $M$ is a subspace, so it's closed under linear combinations.