I have a random walk $X_n$ started at $0$ with jump distribution supported on $\mathbb Z_{\leq 1}$, centered and with finite variance.
I want to show the following monotonicity property for $n$ big enough:
Let $a_n$ be a sequence of positive integers such that $a_n\to\infty$ and $a_n<n/2$. Let $c>0$ be a fixed constant. Then, for $z\in\mathbb Z_{\geq 0}$ big enough,
$$\mathbb P (X_i>c, \forall i \in[a_n,n-a_n]|X_i\geq 0, \forall i \in[0,n],X_n=z)\\\geq\mathbb P (X_i>c, \forall i \in[a_n,n-a_n]|X_i\geq 0, \forall i \in[0,n],X_n=z-1)$$
The property in my mind is intuitive but I have no idea how to write a formal proof.
Any help?