I'm taking a graduate level course on Stochastic Processes and encountered the following problem in one of our assignments.
$\textbf{Problem:}$ Fill in the details of the of the following construction of Poisson Process and prove that the last claim is true.
Let $\lambda>0$ and $X_1,X_2,\ldots,X_n,\ldots$ be iid $\exp(\lambda)$ sequence defined on common probability space $(\Omega, \mathcal{F},P)$. Let $\tau_0:=0$ and $\tau_n := X_1+X_2+\cdots+X_n$ for each $n\geqslant 1$. Define a family $\{N_t:t\geqslant 0\}$ of random variables by $N_0:=0$, $N_t(\omega):=\max\{n\in\mathbb{N} : \tau_n(\omega) \leqslant t\}$ for all $\omega\in\Omega, t>0$.
Claim: Each $N_t$ takes value in $\mathbb{N}$ with probability 1 and $(N_t)$ is a Poisson Process.
Now, I don't understand what does the question mean by "fill in the details". I have seen the construction of Poisson Process from point processes but I cannot think of how to apply the same methods to this question.