Given a closed convex subset $M$ of a Hilbert space $H$, let $x \in H$ be a point s.t. it is not in $M$, i.e. $x \not \in M$.
Let $d = \text{inf}_{y \in M}||x-y||$, where $||.||$ is the inner product and $d$ is the smallest distance from $x$ to the unique point $y$ in $M$..
The text am reading says that there is a sequence of vectors $y_n \in M$ s.t. $||x-y_n|| \rightarrow d$.
Now, it says that using the convexity of $M$ and the parallelogram law, we can show that $y_n$ is a Cauchy sequence, i.e.
$$ ||y_n-y_m||^2 = 2||y_n-x||^2 + 2||y_m-x||^2 - ||(y_n+y_m)-2x||^2 \rightarrow 0 \quad \text{as $n,m \rightarrow \infty$} $$
But I could not figure out how the above formula was derived....
how does one apply (1) convexity of $M$ and (2) the parallelogram law to get this formula?
The parallelogram law is $$ 2||x||^2 + 2||y||^2 = ||x+y||^2 + ||x-y||^2.$$ So $$ 2||y_n-x||^2 + 2||y_m-x||^2 = ||(y_n+y_m)-2x||^2+||y_n-y_m||^2.$$