How do I show that $\sum\limits^\infty_{n=n_0+1}(\frac{n_0}{n})^s$ vanishes as $s\to\infty$? This is my last step of the proof but I'm not sure if this is true nor prove it. Could you give me any suggestions?
 A: In your problem, the convergence "as $s \to \infty$" really should be "as ${\rm Re}(s) \to \infty$" to force the point $s$ to move to the right.
You don't want $s \to \infty$ along a vertical line, for example.
You write in the comments about knowing uniform convergence of the series only in compact subsets.  Dirichlet series actually converge uniformly on wider regions than compact subsets. If a Dirichlet series converges at a point $s_0$ (I don't believe boundedness of its partial sums at $s_0$  suffices for what I say next) then it converges uniformly on subsets of the right half-plane ${\rm Re}(s) > {\rm Re}(s_0)$ that form an angular sector opening to the right out of $s_0$ at less than a $180$-degree angle. This looks like the region at $0$ opening up to the right between the lines $y = mx$ and $y = -mx$ for $x > 0$ and some (finite) $m$.
Such an angular region is called a Stolz angle.  Having $s \to \infty$ within a Stolz angle is more generous than $s \to \infty$ along a horizontal line (${\rm Re}(s) \to \infty$ while ${\rm Im}(s)$ is fixed).  What Stolz angles avoid is having $s \to \infty$ in a way that allows $|s|$ to grow a lot faster than $|{\rm Re}(s)|$ (e.g., it avoids  $s \to \infty$ along a vertical line).
You can find pictures of Stolz angles by a Google image search on "Stolz angle", but the images you'll see are Stolz angles opening up into a disc from a point on the boundary. Stolz (in 1875) introduced "his" regions in order to generalize Abel's theorem for power series from radial convergence at a boundary point of the disc of convergence of a power series to larger subsets of that disc.  Later on, this concept was applied to Dirichlet series. For a Dirichlet series, the significance of Stolz angles at a point $s_0$ is that inside such a region the ratio $|s - s_0|/{\rm Re}(s-s_0)$ is bounded above.
A book that proves a Dirichlet series converges uniformly inside a Stolz angle (opening up to the right) is Serre's Course in Arithmetic. See Proposition 6 of Chapter VI.  The Stolz angle condition occurs in the proof where Serre writes  "$|z|/R(z) \leq k$".  His $R(z)$ means ${\rm Re}(z)$.
This business with Stolz angles isn't too important.  If a Dirichlet series converges at $s_0$ then it converges absolutely for ${\rm Re}(s) > {\rm Re}(s_0) + 1$, so if you are interested in behavior of a Dirichlet series as ${\rm Re}(s) \to \infty$ then you can focus on a half-plane of absolute convergence, in which case ${\rm Im}(s)$ is irrelevant to convergence issues: a Dirichlet series that converges absolutely at a point $s_0$ converges uniformly on the entire half-plane ${\rm Re}(s) \geq {\rm Re}(s_0)$.
A: $$\sum\limits^\infty_{n=n_0+1}\left(\frac{n_0}{n}\right)^s=\sum\limits^\infty_{n=1}\left(\frac{n_0}{n}\right)^s-\sum\limits^{n_0}_{n=1}\left(\frac{n_0}{n}\right)^s=\zeta (s)-1$$
For large values of $s$
$$\zeta (s)-1=\sum_{k=2}^\infty k^{-s}\sim 2^{-s}$$
