Let $\left( X_n \right)_{n \ge 1}$ be a sequence of iid random variables with discrete distribution $p_k=P(X=k)>0$ for $k=0,1,2,...,N$, where $N<\infty$ and $\sum_{k=0}^Np_k=1$.
We define $T_1=1$ and $T_n=\min \{ k \in \mathbb{N} : k>T_{n-1} \wedge X_k \ge X_{T_{n-1}} \}$ for $n=2,3,...$ and $W_n=X_{T_n}$.
Let $m,s>1$. How can I formally prove that $f(l)=E\left( W_{m+s} | W_{m}=l \right)$ is a strictly increasing function?
Note that $W_n$ is a subsequence of $X_n$ and is increasing