Poisson process and Heaviside function

Question

Show that Poisson process $p(t)$ of intensity $\lambda$ can be written as $$p(t)=\sum_{t>t_n}\delta(t-t_n),$$ where function $\delta:\mathbb{R}\rightarrow\mathbb{R}$ is Heaviside's function: $$\delta(t)=\begin{cases}0 \textrm{ if }t<0\\1 \textrm{ if }t\geq0, \end{cases}$$ and sequence $(t_n)_{n\in\mathbb{N}}$ is sequence of random variables of vaules in $\mathbb{R_+}$ such that increments $$\Delta t_i=t_i-t_{i-1},$$ where $i=1,2\ldots, t_0=0$ are nonnegative, independent and have the same distribution of density $$g(t)=\lambda e^{-\lambda t}$$ for $t\geq0.$