How to show that poisson process with linear drift diverges to $\infty$

Question

If $({N_t})_{t\in\mathbb{R}}$ is a poisson point process at rate $\lambda$, I want to show that $N_t-c t$ diverges to infinity as $t\to\infty$ if $c<\lambda$. I can show using the law of large numbers that for $n\in\mathbb{N}$ we have $\frac{N_n}{n}\to\lambda$ as $n\to\infty$. And from there I can conclude $N_n-cn\to\infty$ as $n\to\infty$. But I don't know how to show this for general $t\in\mathbb{R}$ when $t\to\infty$.

Answer 1

Let $n$ denote $\lfloor t\rfloor$, for ease of reading. Use $$ \frac{N_n}{n+1}\le \frac{N_t}t\le \frac{N_{n+1}}n $$ to show that $N_t/t\to \lambda$ as $t\to\infty$ through the reals.