Show that there exist $A>0$ (not depend on $m$) such that $|u_{m}|>A$

Question

Let $(u_{m})_{m≥1}$ be a strictely positive and convergent sequence. Then show that there exist $A>0$ (not depend on $m$) such that $$|u_{m}|>A$$ for all $m≥1$.

I think to exploite the fact that any convergent sequence is bounded (both above and below). However without any success.

Answer 1

If $(u_n)$ is convergent, then it's bounded. Let denote $M=\inf |x_n|$. Then, $A=M-1$ work.