I am reading my lecture notes on Markov chains, and in the proof of one proposition the following statement is made:
"For $n = 1,2, \dots$ the event $\{n \leq T_k\}$ depends only on $X_0, \dots, X_{n-1}$". Where $T_k$ is the first passage time on $k$. I'm slightly confused about this, because it seems to me that the event also depends on the value of $X_n$. Can someone please point out what I'm missing?