Running maximum = local martingale?

Question

Let $X$ be a Brownian Motion with drift $\mu>0$ and $X^*$ its running maximum. Is $X^*$ a local martingale? If I construct the sequence of stopping times $\tau_k = \inf \{s >\tau_{k-1}, X(s)>X(\tau_{k-1}) \}$ and let $Y(t) = X(t) \mathbb{1} _{t \in [\tau_{k-1},\tau_{k})}$ I am almost there because $Y(t)=X^*(t)$. Is there a better (more direct or more elegant) proof or I am wrong?