Let $(N_t)_{t\ge0}$ be a càdlàg Poisson process on a probability space $(\Omega,\mathcal A,\operatorname P)$ with intensity $\lambda\ge0$, $\tau_0:=0$ and$^1$ $$\tau_k:=\inf\left\{t>\tau_{k-1}:\Delta N_t>0\right\}$$ for $k\in\mathbb N$.
How can we show that $$N_{\tau_k}=k\;\;\;\text{almost surely}\tag1$$ for all $k\in\mathbb N$? Moreover, I would like to show that $$N_t-N_s=\left|\left\{r\in[s,t]:N_{r-}\ne N_r\right\}\right|\tag2$$ for all $t\ge s\ge0$ almost surely.
Intuitively, the claims seem to be trivial, but how can we prove it? I was able to show that $N$ is almost surely nondecreasing and $$\operatorname P\left[\forall t\ge0:\Delta N_t\in\{0,1\}\right]=1\tag3$$ from which the claims should actually easily follow.
$^1$ As usual, $\Delta f(t):=f(t)-f(t-)$, $f(t-):=\lim_{s\to t-}f(s)$.