In the space $\ell^1(\Bbb R)=\{x=(x_k)_{k=1}^\infty;x_k\in\Bbb R\text{ and }\|x\|=\sum_{k=1}^\infty|x_k|\}$ consider, for a fixed $y\in\ell^1$, the sets $B=\{x\in\ell^1;\|x\|\le 1\}$ and $M=\{x\in\ell^1;|x_k|\le|y_k|\}$. Find a continuous, real-valued function on $\ell^1$ that does not have a largest value on $B$ and show that $M$ is compact subset of $\ell^1$.
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I got a hint that for the compactness of M I need to show that every row in B has a convergent subrow (diagonal argument) but I don't know how to show this, does anybody know how to show this?