Consider $M_t:=\sup\limits_{t\geq0}B_t$. I have managed to prove that the law of $M_t$ is concentrated on $\{0,\infty\}$ using the distribution of scaled Brownian motion.
However, I am asked to prove $$\mathbb{P}(M_t=0)\leq\mathbb{P}\left(B_1\leq0,\sup_{t\geq0}B_{t+1}−B_1 =0\right)$$ to show that $\mathbb{P}(M_t=0)=0$, and somehow show that the BM path is not differentiable at zero a.s. Where does the inequality come from? The joint distribution suggests reflection principle, but as with most reflection principle problems I have asked before I am unable to see how to begin; I have seen other questions on this site and they don't seem to use this inequality. Furthermore how does this lead to the subsequent requested results? Thanks!