Conditional probability branching process Consider a discrete time branching process $X_{n}$ with $X_{0}=1.$ Establish the simple inequality $$P\{X_{n}>L\ \textrm{for some}\ 0\leq n\leq m\ |\ X_{m}=0 \}\leq [P\{X_{m}=0\}]^L$$
Note: This issue is the second edition of the book "The First Course in Stochastic Processes" Samuel Karlin author. Could anyone help me to solve it? I have no idea to answer it! I thank you!
 A: Let $T:=\inf\{n:X_n>L\}$. Also define $\mathsf{E}_x[\,\cdot\,]:=\mathsf{E}[\,\cdot\mid X_0=x]$, $\mathcal{F}_n:=\sigma(X_0,\dots,X_n)$, and $\mathcal{F}_T:=\left\{A:A\cap\{T=n\}\in\mathcal{F}_n\text{ for all } n\right\}$.
Then since 
$$
\{X_n>L\text{ for some }0\le n\le m\}=\left\{\max_{0\le n\le m}X_n>L\right\}=\{T\le m\}
$$
we have
\begin{align}
&\mathsf{P}_1\!\left(\max_{0\le n\le m}X_n>L,X_m=0\right)=\mathsf{P}_1(T\le m,X_m=0) \\
&\qquad=\mathsf{E}_1\!\left[1\{T\le m\}1\{X_m=0\}\right] \\[0.6em]
&\qquad=\mathsf{E}_1\!\left[1\{T\le m\}\mathsf{E}_1[1\{X_{(m-T)+T}=0\}\mid \mathcal{F}_T]\right] \\[0.3em]
&\qquad\overset{(1)}{=}\mathsf{E}_1\!\left[1\{T\le m\}\mathsf{E}_{X_T}1\{X_{m-T}=0\}\right] \\[0.3em]
&\qquad\overset{(2)}{=}\mathsf{E}_1\!\left[1\{T\le m\}(\mathsf{P}_1(X_{m-T}=0))^{X_T}\right] \\
&\qquad\overset{(3)}{\le} \mathsf{E}_1\!\left[1\{T\le m\}(\mathsf{P}_1(X_m=0))^{L+1}\right]\le (\mathsf{P}_1(X_m=0))^{L+1},
\end{align}
where $(1)$ holds by the strong Markov property, $(2)$ follows from the fact that $\mathsf{P}_x(X_k=0)=(\mathsf{P}_1(X_k=0))^{x}$, and $(3)$ holds because $X_T\ge L+1$ and $\mathsf{P}(X_k=0)$ is non-decreasing in $k$.
Consequently,
$$
\mathsf{P}_1\!\left(\max_{0\le n\le m}X_n>L\mid X_m=0\right)=\frac{\mathsf{P}_1(\max_{0\le n\le m}X_n>L,X_m=0)}{\mathsf{P}_1(X_m=0)}\le (\mathsf{P}_1(X_m=0))^L.
$$
