The problem is:
Let E be a Banach space and $F\subset E$ be a closed linear subspace. Prove that for every $x \in E$ there exists $y \in F$ such that $\left\Vert x-y\right\Vert =\inf\left\{ \left\Vert x-z\right\Vert :\, z\in F\right\} =\left\Vert x+F\right\Vert _{E/F}$.
My efforts:
By definition, we know that $\left\Vert x+F\right\Vert _{E/F}:=\inf\left\{ \left\Vert x+z\right\Vert :\, z\in F\right\} $, and since $F$ is a linear subspace, we have that $\inf\left\{ \left\Vert x+z\right\Vert :\, z\in F\right\} =\inf\left\{ \left\Vert x-z\right\Vert :\, z\in F\right\} $.
My idea:
We know that for every bounded sequence in a reflexive Banach space, there exists a weakly convergent subsequence.
My question:
How can I now use the weak convergence to prove the first equality of my statement? Have I done any mistake so far?