Convergent Sequence in Normed Vector Space is Weakly Convergent

Theorem
Let $\struct {X, \norm \cdot}$ be a normed vector space.

Let $x \in X$.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$ converging to $x$.

Then $\sequence {x_n}_{n \mathop \in \N}$ converges weakly to $x$.