I'm completing a proof about constructing a Poisson Process, let $X_i \sim \exp(\lambda)$ for $i =1, \ldots n$ (all independent from each other). I've shown that their sum $S_n = \sum_{i=1}^n X_i$ is a $\Gamma(n,\lambda)$, in particular if $F_n$ is the distribution of $S_n$ then integrating by parts $F_n$ we can get the following relation:
$$F_{n+1}(t) = F_n(t) - \frac{(\lambda x)^{n} e^{-\lambda x}}{n!} $$.
Finally, if $N(t) = \max\{n: S_n\leq t\}$ then I want to prove $N(t) \sim \operatorname{Poisson}(\lambda t)$. It's easy to check that $\{ N(t) = n \}$ is equivalent to $\{S_n \leq t, S_{n+1}>t \}$ and also $\{N(t) \geq n \}$ it's equivalent to $\{S_n \leq t \}$. My question is, why $$P(N(t)=n) = P(S_{n+1}>t) - P(S_n>t)$$ holds?
From the previous, $P(N(t) = n) = P(S_n \leq t, S_{n+1}>t)$ then I don't know what to do.