Let $(f_n)$ be a bounded sequence of bounded linear functionals on a real normed linear space $X$. Then there exits a linear functional $f\in X^*$ such that $\liminf\limits_{n\to \infty}f_n(x)\leq f(x)\leq \limsup\limits_{n\to \infty}f_n(x)$ for all $x\in X$.
I proceed as follows: Since for each $x\in X$, $(f_n(x))$ is a bounded real sequence, by Bolzano-Weierstrass theorem there exists a subsequence $(f_{n_k})$ of $(f_n)$ such that $(f_{n_k}(x))$ converges. We define $f(x)=\lim\limits_{k\to \infty}f_{n_k}(x)$ for all $x\in X$. The very first problem I face is in showing $f$ to be linear. Once it can be shown to be linear, the problem is solved. Any hint is appreciated.