= Solution
Put $d=\inf\{\|x\|:x\in C\}$. 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 $n\ge1$, 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 $\|x_0\|\le d+1/n$ for every $n$, so
$$
\boxed{x_0\in C,\qquad\|x_0\|=\min_{x\in C}\|x\|=d.}
$$
This 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>.
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