What does it mean to say that $\left(X_{t}\right)$ is a renewal process?

Let $\left(X_{t}\right)$ be a renewal process with holding times $S_{1}, S_{2}, \ldots$ and let $s>0$. For $n \geqslant 1$, set $T_{n}=S_{X_{s}+n}$. Show that

$$\mathbb{P}\left(T_{n}>t\right) \geqslant \mathbb{P}\left(S_{n}>t\right), \quad t \geqslant 0,$$

for all $n$, with equality if $n \geqslant 2$.

Consider now the case where $S_{1}, S_{2}, \ldots$ are exponential random variables. Show that

$$\mathbb{P}\left(T_{1}>t\right)>\mathbb{P}\left(S_{1}>t\right), \quad t>0$$

and that, as $s \rightarrow \infty$,

$$\mathbb{P}\left(T_{1}>t\right) \rightarrow \mathbb{P}\left(S_{1}+S_{2}>t\right), \quad t \geqslant 0$$