let us consider Poisson process $N_t$ with $\lambda$ parameter and a stopping time $T=\inf\{t\ge 0;\,N_t=a\}$, where $a\in\mathbb{N}$. I would like to show that $ET=\frac{a}{\lambda}$, so I want to use Doob Theorem. I want to show that $P(T<\infty)=1$. What I have: $P(T=\infty)\le P(N_t<a,\,t\ge 0)\le P(N_t<a)=e^{-\lambda t}\sum_{k=0}^{a-1}\frac{(\lambda t)^k}{k!}\ldots$
I want to make some boundary for this sum independent of $t$ (don't know if it is even possible) and pass to the limit with $t\to\infty$. Any ideas?