I'd like to prove by counterexample that there is no norm on the space of all real-valued bounded sequences such that $\lim_{n \to \infty} \|f_n - f\| = 0$ is equivalent to $\lim_{n \to \infty} f_n = f$. In other words, I want to find a sequence $(f_n)_{n\in \mathbb N} \subset \ell^\infty(\mathbb R)$ which converges to $f$ (pointwise) but for which $$ \lim_{n \to \infty} \|f_n - f\| \neq 0. $$
This question is inspired by the paper A Note on Pointwise Convergence, which proves that pointwise convergence is not "metrizable" on $\mathcal{C}[0,1].$ I haven't been able to find any proofs that this result holds in the space of bounded sequences, but since the space is not even complete under any norm by Baire, I'm quite sure a counterexample exists. I'd really appreciate any ideas y'all have.
Let $\lVert \bullet \rVert$ be any norm on $\ell^\infty$. For each $i \in \mathbb{N}$, let $\delta_i$ be the Kronecker delta sequence (that is, $\delta_i(j) = 1$ if $i = j$, and $\delta_i(j) = 0$ if $i \neq j$).
Define $f_i = \frac{1}{\lVert \delta_i \rVert} \delta_i$, so that $\lVert f_i \rVert = 1$ for all $i$. Then $f_i \to 0$ pointwise, but
$$\lVert f_i - 0 \rVert = 1$$
does not converge to $0$.