I am trying to solve the exercises 11.3 from the book Le Gall: Measure Theory, Probability and stochastic processes.
Exercise 11.3: Let $(Xn)$ be a sequence of independent real random variables uniformly distributed over $[0,1]$. Define the record times of the sequence by
$T_1 = 1$ and $T_p := inf ${$n > T_{p-1} : X_n > X_{T_{p-1}}$ } for $p \geq 2$,with the convention of inf{$\emptyset$} = $\infty$. Show that $P(T_p < \infty) = 1$ for every $p \in N$. Then determine the law of $T_2$, and prove that, for every $p \geq 2$ and $k \in N$,
$E[1_{{T_p = k}} | (T_1, ..., T_{p-1} )] = E[1_{{T_p = k}} | T_{p-1}] = \frac{T_{p-1}}{k(k-1)} 1_{{k>T_{p-1}}}$.
I am already stuck by showing that $P(T_p < \infty) = 1$. I started with$P(T_p < \infty) = 1 - P(T_p = \infty)$ and $T_p = \infty$ if there is no $n>T_{p-1}$ such that $X_n > X_{T_{p-1}}$ which means that for all $n > T_{p-1}$ it holds that $X_n \leq X_{T_{p-1}}$.
I am thankful for any help:)