Weakly convergent sequence is bounded
= Weakly convergent sequence is bounded
Every <weakly convergent> sequence in a <normed vector space> is a <bounded sequence>. Apply the <Uniform boundedness principle> on the Banach space $X^*$ to the evaluation maps $Jx_n:f\mapsto f(x_n)$. They are pointwise bounded, while the <canonical embedding into the bidual> gives $\lVert Jx_n\rVert=\lVert x_n\rVert$.