How to prove that width of general dirichlet series is given by this

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My instructor gave me this question for the assignment and I am not able to solve this.

Prove that $ 0\leq \sigma_{a} -\sigma_{c} \leq L$, where $\displaystyle L=\limsup_{ n\to \infty} \frac{\log n} {\lambda_n}$.

Here is some information about the general Dirichlet series:https://en.m.wikipedia.org/wiki/General_Dirichlet_series

So, I tried to use the formula of abscissa of convergence and abscissa of absolute convergence but that doesn't seem the right thing to do here.

So, can you please outline a proof as assignment might not be discussed due to ongoing pendamic situation.

Thank you.