Distribution of random walk changes

Question

Say that $R_t = R_{t-1}+e_t$, where $e \overset{i.i.d.}\sim N(o, \sigma^2)$. Take now $\Delta R_t = e_t$, which is the differenced series. Say now that $S_t$ is equal to 1 whenever $ \Delta R_t \geq 0$ and - 1 otherwise. Can we argue that $S_t \overset{i.i.d.} \sim Bernoulli(p) $? Where $p=P(S_t=1)$. To me it seems reasonable given the definition of $\Delta R_t$, but could you provide a rigorous proof or disproof? Moreover, is it correct to include the case $\Delta R_t = 0$ with $\Delta R_t > 0$? It seems correct to me since virtually $P(\Delta R_t = 0) = 0$. Thank you.