Why a continuous Markov chain is recurrent iff its embedded jump chain is recurrent?

Question

I read this statement in the proof of this theorem 5 from the website Random Services. I am a little confused and I didn't find proof for it. Thanks for any explanation or clue!

Answer 1

Define $S_x = \inf\{t: X_t = x, X_0 = x\}$ where $S_x$ is the minimum amount of time it takes to first revisit state $x$ (i.e. if it is never revisited $S_x = \infty$) and assume that the jump chain $\{X_n\}$ is recurrent for $x$.

Notice that $S_x$ is the sum of the holding times for all states, not $x$. That is to say: $$ S_x = \sum_{n = 0}^{\tau_x - 1} T_n $$ where each $T_k$ is the holding time for the visited state at the $k$-th step and $\tau_x$ represents the number of steps it takes to revisit state $x$ for the first time.

Since we assumed that the jump chain is recurrent for $x \Rightarrow \tau_x < \infty \Rightarrow T_k < \infty$. In other words, $S_x$ itself must also be finite, so $x$ must be recurrent for the chain $\{X_t\}$.