Computing the PMF for N(t) in a renewal process

Question

I'm in a stochastic processes course, and we just started on renewal theory. Unfortunately, we skipped the section on queuing theory, and nearly example in my textbook for renewal processes uses examples from the queuing theory chapter.
That being said, I think I have a moderate understanding of this material, and am looking for a push in the right direction for a problem from Kulkarni's Modeling and Analysis of Stochastic Systems.
Problem Text
Let $\{N(t), t\geq0\}$ be a renewal process with iid inter-renewal times with common pmf
$P(X_{n} = i) = \alpha^{i-1}(1-\alpha),$ $i \geq 1$, where $0 \lt \alpha \lt 1$.
Compute $P(N(t) = k)$ for $k = 0,1,2,...$.

Relevant Theorems
A renewal process generated by a sequence of iid random variables $\{X_{n},n\geq1\}$ with common cdf $G(*)$ is completely characterized by $G(*)$.

Let $G_{k}(t) = P(S_{k}\leq t), t\geq 0$.
Then $p_{k}(t) = P(N(t) = k) = G_{k}(t) - G_{k+1}(t), t \geq 0$.

So, I'm trying to connect the dots between the PMF for $X_{n}$ and this $G_{k}(t)$ function. I know that for renewal processes $X_{n} = S_{n} - S_{n-1}$, and thus represents time between events $n-1$ and $n$. I integrated the PMF in order to get the CDF, which is $P(X_{n} \leq i)$. However, after this point, I'm not seeing any link between the CDF and this $G(*)$ function the theorem cites. The theorem claims $G(*)$ is the common CDF of $X_{n}$, which I've found, but then claims $G_{k}(t) = P(S_{k}\leq t)$. I don't see how both of those statements can be true. If $G(*)$ is the CDF for $X_{n}$ then $G(*) = P(X_n \leq t) = P((S_n - S_{n-1}) \leq t)$ and this clearly does not equal $P(S_{k}\leq t)$.

So, I'm quite certain that my understanding of these functions is not correct. Otherwise the book is wrong, and, well, I trust the textbook author over me 99.99% of the time.

Any clarification on where I went wrong in my application of the theorems or insight on the problem would be immensely helpful. I'd prefer no one directly answer the problem, as I think working through it with some hints will teach me a lot more.

Thanks!

Answer 1

The theorem claims [that] $G$ is the common CDF of $X_{n}$ (...) but then claims [that] $G_{k}(t) = P(S_{k}\leq t)$. I don't see how both of those statements can be true. If $G$ is the CDF [of every] $X_{n}$ then $G(t) = P(X_n \leq t) = P(S_n - S_{n-1} \leq t)$ and this clearly does not equal $P(S_{k}\leq t)$.

Why should it? Nobody said that $G=G_k$ and actually $G=G_1$ but $G\ne G_k$ for every $k\geqslant2$.

Note that the theorem neither claims that $G$ is the common CDF of $X_{n}$ nor that $G_k$ is the CDF of $S_k$, these are assumptions of the theorem.