= Solution
Let $S_1,S_2,\ldots$ be independent <exponential random variables> of rate $\lambda$, let $T_n=S_1+\cdots+S_n$ and $T_0=0$, and define $N_t=\max\{n:T_n\le t\}$. The <strong law of large numbers> gives $T_n/n\to1/\lambda$, so there is no finite-time explosion. This defines the <Poisson process> of rate $\lambda$.
For $n\ge1$, repeated <convolution> gives $T_n$ the <gamma distribution> with density $\lambda^n s^{n-1}e^{-\lambda s}/(n-1)!$. Independence of $S_{n+1}$ gives
$$
\mathbb P(N_t=n)=\int_0^t\frac{\lambda^n s^{n-1}e^{-\lambda s}}{(n-1)!}e^{-\lambda(t-s)}ds=e^{-\lambda t}\frac{(\lambda t)^n}{n!}.
$$
For $n=0$, the probability is $\mathbb P(S_1>t)=e^{-\lambda t}$. Hence $\boxed{N_t\sim\operatorname{Poisson}(\lambda t)}$ for all $t\ge0$, including the degenerate law at $t=0$.
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