Consider a random walk starting at $x_0 \in [0,1]$. At any discrete time step $k \in \mathbb{N}$, the new position is given by: $$x_k = \min\{\max\{x_{k-1}+C_k,0\},1\}$$ where $C_k$ are continuous i.i.d. random variables with support in $\mathbb{R}$. In other words, we are moving at random in $[0,1]$ and we have a certain probability to be ''absorbed'' at $0$ or $1$ and then eventually reflected back.
I am trying to prove that the average minimum time needed to hit state $0$ for the first time (hitting time) starting from a generic state $x_0=y \in [0,1]$, call it $\mathbb{E}_y[\tau_0]$, is always smaller or equal to the same hitting time but starting from state $x_0=1$, call this $\mathbb{E}_1[\tau_0]$. In other words, I need to show that $$ \mathbb{E}_y[\tau_0] \le \mathbb{E}_1[\tau_0]$$ This seems obvious to me as the state $y$ will always be closer to state $0$ than state $1$, and hence it should take a shorter time to hit state $0$.
Any idea about how this could be formally proved?