Random walk on a segment with infinite time

Question

Given a point particle on a segment $L$ of length $1$, $(L=[0,1])$, assume the particle moving randomly in such a way: $p_{(k+1)}=p_k+\delta_k$ where $p_{k+1}$ is the position on the segment at time $t_{k+1}$ and $\delta_k$ a random number $\delta_k\in\mathbb{R},-1\lt\delta_k\lt1$. If $p_k+\delta_k\lt 0,p_{k+1}=|p_k+\delta_k|$ and if $p_k+\delta_k\gt1,p_{k+1}=p_k+\delta_k-1$ (reflecting walls), given infinite time, can we prove the particle will touch every point in the segment? If the set of all the time intervals is uncountable, I suppose the answer could be 'yes'. Is it a good answer? Thanks.