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?