Increments in the Poisson process

Question

We have a sequence of i.i.d. exponential random variables $X_n$, of parameter $c>0$. Define the partial sum as $S_n = \sum_{k=1}^n X_k$, and, for $t \ge 0$, the process $N_t = \sum \cal I_{S_n <t}$, where $\cal I$ is the indicator function. Show that $N$ is indeed a Poisson process, which translates in showing that increments are independent, and their law is a Poisson r.v. of parameter $c$.

I think i figured out that increments are independent, just by computing the expected value of the product and using independence of $X$. I am now stuck in proving their law is Poisson. Can someone help?