Uniform bound from pointwise absolute summability of dual evaluations

Let $(x_n)$ lie in a normed space $X$ and suppose ${\sum }_n|f(x_n)|<\infty$ for every $f\in X^{\ast }$. Then some $C$ satisfiesApply the uniform boundedness principle on the Banach space $X^{\ast }$ to the finite signed sums $f\mapsto {\sum }_n{a}_{n}f(x_n)$ with $|a_n|\le 1$.