# Transforming sum of n exponential distribution to a poisson distribution

Let $$X_1,…,X_n$$ be i.i.d exponential random variable with mean $$\lambda$$

$$S=X_1+…+X_n$$

So by finding the mgf of S, we get that $$S \sim Gamma(n,\lambda)$$

The problem I am stuck at is how can we show:

for a given positive t, $$N = max\{n:n\geq 1, S \leq t\}$$ has a Poisson distribution

how should I start?