I'm quoting from page 261 of "Measure theory and probability theory" by Krishna and Soumendra.
Let $N(t)$ and $S(t)$ be the associated renewal process and renewal sequence, the book proves the following
\begin{gather*} P (N (t) > k) ≤ C(t)\lambda^k \ \ \ \forall t \end{gather*}
where $C(t)$ is a positive constant depending on $t$ and $\lambda \in(0,1)$ not depending on $t$. The proof is quite short:
\begin{gather*} P(N(t)<k)=P(S_k \leq t) \\ =P(e^{-\theta S_k} \geq e^{-\theta k}) \ \ \ \text{for} \ \ \theta>0 \\ \leq e^{\theta t}E(e^{-\theta S_k}) \ \ \ \ \text{by Markov's inequality} \\ = e^{\theta t}E(e^{-\theta X_0})\left(e^{\theta t}E(e^{-\theta X_1})\right)^k \end{gather*} and as $\theta$ goes to infinity we get $E(e^{-\theta X_1}) \rightarrow P(X_1=0)$ that is less then 1 by hypothesis.
Then it says that from this follows that $P(N(t)=k)= O(\lambda^k)$, but for me it's not clear how to prove this implication. May you demystify it? Thanks.