Closed convex subset of $l^1$ has infinitely many elements of minimal norm

Question

I need help with the following:

Closed convex subset $M=\{x \in l^1: \sum_{n=1}^{\infty}x_n=1\}$ of space $l^1$ containes infinitely many elements of minimal norm. Prove this statement.

I wanted to find $\underset{x \in M}{inf ||x||}$, but I only see that $$||x||= \sum_{n=1}^{\infty}|x_n| \geq \sum_{n=1}^{\infty}x_n=1,$$ so inf is $\geq 1$. I don't know how to find inf, but I don't know also what to do with the other part.

Thanks a lot!