Is the set of decreasing sequences on the unit ball of $l^1[0,\infty)$ a compact set under the sup norm? Formally, let
$$
\left\{\boldsymbol{x} = (x_1,x_2,x_3,\ldots)\in [0,1]^{\mathbb{N}^*} : 1 \geq x_1 \geq x_2 \geq \ldots \geq 0 ~; \sum_{i=1}^{\infty} x_i \leq 1\right\}.
$$
Is this a compact set with the sup norm $\|\boldsymbol{x}\|_{\infty} = \sup_{i\in \mathbb{N}} |x_i|$ ?

Consider a sequence $y_n$,
$$y_n=(\frac1n,\frac1n,\frac1n,\cdots,\frac1n,0,0,\cdots).$$
