Is this argument sufficient to show alternating zeta series converges for $Re(s)>0$?

Question

The alternating zeta series

$$ \eta(s) = \sum\frac{(-1)^{n+1}}{n^s} $$

is known to converge for $\sigma=\Re(s)>0$.

Question: There are many proofs offered but I wondered if the following simple (suitable for students pre-university) argument is correct and sufficient?

1. It is known that Dirichlet series have an abscissa of convergence $\sigma_c$, where the series converge for $\sigma>\sigma_c$.

2. This means we only need to find the convergence interval in the real domain.

3. In the real domain only, the alternating zeta series $\eta(\sigma)$, converges for $\sigma>0$.

4. Therefore, the alternating zeta series $\eta(s)$ converges for $\Re(s)>0$ in the complain domain.