I'm a little curious to understand the $||\cdot ||_{\infty}$ and $||\cdot ||_{2}$ norms a little better.
The following example presents a sequence $(x_n)_{n \in \mathbb N}$ in $\ell^2$ that is in the unit ball, but $|| x_n ||_{\infty} \to 0$, as $n \to \infty$.
More specifically, the first $n$ elements (or coordinates) of $x_n$ are equal to $1/\sqrt{n}$. So, we can consider this example a little "simple".
I would like to know if we can construct a sequence $(x_n)$ such that $$||x_n ||_{2}> r, \,\, \forall n \in \mathbb N, \quad r>0$$ and $|| x_n ||_{\infty} \to 0$, as $n \to \infty$, but the non-null coordinates $x_{jn}$'s that make up $x_n$ are not all the same?
Does the number of non-zero elements $x_{jn}$ necessarily have to be finite or can we have infinitely many non-zero coordinates? (Remember that we necessarily have to have $|x_{jn}|^2 \to 0$, as $j \to \infty$ for all $n$).
Letting $x_{jn} = \sqrt{\frac{1}{n + 1}(\frac{n}{n + 1})^{j - 1}}$ gives you an example. $||x_n||_2 = 1$ for all $n$ but $||x_n||_\infty = \sqrt{\frac{1}{n + 1}} \rightarrow 0$.