Existence of Abscissa of Convergence/General

Theorem
Let $s = \sigma + i t$

Let $\displaystyle f \left({s}\right) = \sum_{n \mathop = 1}^\infty a_n e^{-\lambda_ns}$ be a general Dirichlet series.

Then there exists a extended real number, $\sigma_0$, such that
 * $(1): \quad$ For $\sigma<\sigma_0$, $\displaystyle f \left({s}\right)$ diverges
 * $(2): \quad$ For $\sigma>\sigma_0$, $\displaystyle f \left({s}\right)$ converges.

Proof
If there does not exist an $s_0$ such that $\displaystyle f \left({s_0}\right)$ converges, then $\sigma_0 = \infty$ and the theorem is vacuously true

If there does exist such an $s_0$, let $\sigma_0$ be the infimum of the real part of all such $s_0$, where $\sigma_0= -\infty$ if the set is not bounded from below.
 * It is clear that $f \left({s}\right)$ diverges if $\sigma<\sigma_0$ by construction of $\sigma_0$ as the infimum of all convergent s


 * By Dirichlet Series Convergence Lemma, We have $f \left({s}\right)$ converges for all s such that $\sigma>\sigma_0$