How may I proceed with this problem: Show directly from the infinitesimal definition of Poisson process that the first jump time $J_1$ of a Poisson process of rate $\lambda$ has exponential distribution of parameter $J_1$.
I have this definition: Let $(X_t){t\geq 0}$ be an increasing, right-continuous integer-valued process starting from $0$. Let $0<\lambda<\infty$. $(X_t){t\geq 0}$ has independent increments and, as $h\downarrow 0$, uniformly in t,
$\mathbb{P}(X_{t+h}-X_t=0)=1-\lambda h +o(h)$,
$\mathbb{P}(X_{t+h}-X_t=1)=\lambda h +o(h)$