It is known that $\|\mathbf x\|_\infty := \displaystyle\lim_{p\to\infty}\left(\sum_{k=1}^n |x_k|^p\right)^\frac 1 p = \max(|x_1|, \ldots, |x_n|)$ where $\mathbf x = (x_1, \ldots, x_n) \in \mathbb R^n.$ But is this true for infinite vectors as well?
Define $\|\mathbf y\|_\infty = ||(y_1, y_2, y_3, \ldots)||_\infty := \displaystyle\lim_{\min(k,p) \to \infty} \left( \sum_{n=1}^k |y_k|^p \right)^\frac 1 p$. Given that the real valued sequence $\{ y_k \}_{k=1}^\infty$ is bounded from above, will $\|\mathbf y\|_\infty = \max(|y_1|, |y_2|, |y_3|, \ldots)$?
