Convergence of sequences in $l^{\infty}$

Question

I am looking for a sequence of sequences in $l^{\infty}$ (space of bounded sequences) that converges pointwise to the 0 sequence but not uniformly. I have come up with sequences that do that however they don't seem to be bounded, or I have found bounded ones that have both forms of convergence.

Answer 1

Consider the sequence $(e_n)_n$ in $\ell^\infty$ given by$$e_n = (\underbrace{0, \ldots, 0}_{n-1},1,0, 0, \ldots)$$

Clearly $e_n \to 0$ pointwise but $$\|e_n - 0\|_\infty = \|e_n\|_\infty = 1$$ so $e_n \not\to 0$ uniformly.