In this article, at the end of page 11, there is a proof that for a Brownian motion, almost surely, there exist disjoint intervals with the same maximum. The proof uses the random variables $(T_{a,n})_{n \geq 0}$ with $T_{a,n}$ being the $n$-th time the process hits $a > 0$. Since in between $T_{a,n}$ and $T_{a,n+1}$ the process is either below or above $a$ with equal probability, there must exist, almost surely, an interval $[T_{a,n},T_{a,n+1}]$ where the process will be below $a$ and its maximum there will be $a$ as it is on $[0,T_{a,1}]$.
My question is: how are the variables $T_{a,n}$ defined ? A logical definition would be $T_{a,1} = \inf \{t>0 : B_t = a\}$ for $n=1$ and $T_{a,n} = \inf \{t>T_{a,n-1} : B_t = a \}$ for $n \geq 2$. But then, once the process visits $a$, by the Markov property, it revisits it infinitely many times in $[T_{a,1},\epsilon)$ for every $\epsilon$ so it seems that the definition of $T_{a,n}$ does not make sense ($T_{a,n}$ would be the same as $T_{a,1}$ for every $n$). Any explanation will be appreciated.