Question regarding stationary distributions in Probability and random process by Grimmett and Stirzaker.

Question

One page 228, they defined $\rho_i(k)=E(N_i|X_0=k)$ where $N_i=\sum 1_{\{X_n=i\}\cap \{T_k\geq n\}}$. But they proceeded to say that $\rho_i(k)=\sum P(\{X_n=i\}\cap \{T_k\geq n\}|X_0=k)$. I don't see how the latter is the same as the former. I tried to show that $\sum nP(N_i=n|X_0=k)=\sum P(\{X_n=i\}\cap \{T_k\geq n\}|X_0=k)$ but I got stuck. Any suggestions?