sequences and series - Proof of convergence of Dirichlet's Eta Function

I'd like to check directly the convergence of Dirichlet's Eta Function, also known as the Alternating Zeta Function or even Alternating Euler's Zeta Function:

$$\sum_{n=1}^\infty\frac1{n^s}\;\;,\;\;\;s=\sigma+it\;,\;\;\sigma\,,\,t\in\Bbb R\;,\;\;\color{red}{\sigma > 0}.$$

Now, there seems to be a complete ausence of any direct proof of this in the web (at least I didn't find it) that doesn't use the theory of general Dirichlet Series and things like that.

I was thinking of the following direct, more elementary approach:

$$n^{it}=e^{it\log n}:=\cos (t\log n)+i\sin(t\log n)$$

and then we can write

$$\frac1{n^s}=\frac1{n^\sigma n^{it}}=\frac{\cos(t\log n)-i\sin(t\log n)}{n^\sigma}$$

and since a complex sequence converges iff its real and imaginary parts converge, we're left with the real series

$$\sum_{n=1}^\infty\frac{\cos(t\log n)}{n^\sigma}\;\;,\;\;\;\;\sum_{n=1}^\infty\frac{\sin(t\log n)}{n^\sigma}$$

Now, I think it is enough to prove only one of the above two series' convergence, since for example $\;\sin(t\log n)=\cos\left(\frac\pi2-t\log n\right)\;$

...and here I am stuck. It seems obvious both series are alternating but not necessarily
elementwise.

For example, if $\;t=1\;$ , then $\;\cos\log n>0\;,\;for\;\;n=1,2,3,4\;$ , and then $\;\cos\log n<0\;,\;\;for\;\;\;n=5,6,\ldots,23\;$ . This behaviour confuses me, and any help will be much appreciated.