It is a consequence of the more general fact:
If $(X,\Vert \cdot \Vert)$ is a Banach space and $\{x_n\} \subset X$ is a sequence converging weakly to $x$ in $X$, then $\{ \lVert x_n \rVert \}$ is bounded.
Indeed, from the definition of weak convergence we have that $\{\langle f,x_n\rangle\}$ is a bounded subset of $\mathbb{R}$ for every $f \in X'$. Then it suffices to apply the uniform boundedness principle (aka, Banach-Steinhaus theorem) to obtain the result.
References: Proposition 3.5 of Brezis' book Functional Analysis, Sobolev Spaces and Partial Differential Equations or Theorem 1 in chapter V of Yosida's Functional Analysis.