Weakly Convergent Sequence in Normed Vector Space is Bounded

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

Let $\sequence {x_n}_{n \mathop \in \N}$ be a weakly convergent sequence in $X$.

Then $\sequence {x_n}_{n \mathop \in \N}$ is bounded.

Proof
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