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Put $d=\inf \{\Vert x\Vert :x\in C\}$. The nonemptiness of $C$ gives $0\le d<\infty$. By Mazur theorem, a norm-closed convex set is weakly closed. This applies to both $C$ and every closed norm ball.

For $n\ge 1$, the sets $C_n=C\cap (d+1/n)B_X$ are nonempty by the definition of the infimum, weakly closed and nested. They all lie in the weakly compact set $(d+1)B_X$, by the weak compactness characterization of reflexivity. Hence they have the finite intersection property, and compactness gives $x_0\in {\bigcap }_n{C}_n$. Then $x_0\in C$ and $\Vert x_0\Vert \le d+1/n$ for every $n$, soThis proves the norm minimizer in a closed convex subset of a reflexive Banach space assertion without assuming sequential weak compactness. It includes $d=0$; uniqueness is not asserted without an additional condition such as strict convexity of the norm.