Let $\{N_t\}_{t\ge 0}$ a renewal process with intensity $\lambda>0$. Then $\lim_{t\to\infty} \frac{N_t}{t}=\lambda$ a.s.
Here $W_i$ denotes the waiting time of $N_t$ and $T_n:=\sum_{i=1}^n W_i$ and $N_t:=\sup\{n:T_n\le t\}$.
I do not understand one part in its proof.
First we have to show that for any $t\ge0$, $\mathbb P(N_t<\infty)=1.$ For $t=0$ this is clear. Since $\operatorname E(W_1)=\frac 1\lambda$, there exists for every $t>0$ $k_0\in \mathbb N$, such that $\mathbb P(W_1+\dots W_k\le t)<1 $ for every $k\ge k_0$. Thus $$\mathbb P(N_t=\infty)=\mathbb P(T_n < t,\forall n\ge 0) \overset{\text{?}}{\underset{\text{}}{\le}} \mathbb P(W_{k(n-1)+1}+\dots W_{kn}\le t,\forall n\ge 1)=lim_{m\to \infty}(\mathbb P(W_1+\dots +W_k)\le t)^m=0$$
I marked with ? what I do not understand. Why does this hold?
Note that for $n \geq 0$, $$T_{kn} = \sum_{i=1}^{kn} W_i \geq \sum_{i=k(n-1) + 1}^{kn} W_i$$ since $W_i \geq 0$ for each $i$. In particular, we have that $$\{T_j < t \text{ for all } j \geq 0\} \subseteq \{ W_{k(n-1)+1} + \dots + W_{kn} \leq t \text{ for all } n \geq 1\}$$ by considering for each $k$ and $n$ the value $j = kn$. This gives the desired inequality.