Let $\sum_{n=1}^\infty \frac {a_n}{n^x}$:
Prove: if $\sum_{n=1}^\infty \frac {a_n}{n^s}$ converges then $\sum_{n=1}^\infty \frac {a_n}{n^x}$ uniform converges in $[s,\infty)$
My Proof:
For all $x\in [s,\infty)$: $\space$ We can write $\frac{a_n}{n^x}=\frac{a_n}{n^s}\cdot\frac{1}{n^{x-s}}$ and apply Abel's uniform convergence test.
$\frac{1}{n^{x-s}}$ monotonically decreasing and uniformly bounded.
$\sum_{n=1}^\infty \frac {a_n}{n^s}$ uniform converges as it's a series of numbers that converges (given).
$\implies$ $\sum_{n=1}^\infty \frac {a_n}{n^x}$ uniform converges in $[s,\infty)$
Please help me validate this proof. Is it correct? Is it possible to use Abel's test in an interval with infinite endpoint?