Assume $X_1,X_2,\ldots,X_n$ is $n$ record of height of jump of an athelete with i.i.d. distribution.
If we denote the $A_k = \{X_k>\sup_{j<k}X_j\}$ that new record of jump of this athlete appears at time k.
Prove that the current record is independent of future records.That is if $m_1<m_2<m_3<\ldots<m_k$. Then
$P(A_{m_1}\mid(A_{m_2}\cap A_{m_3}\cap\ldots\cap A_{m_k})) = P(A_{m_1}) $
This is an example from Durrett probability fourth edition textbook example 2.3.2.
I can't figure out how to get this result, my attempt is to marginalize the permutation but it fails.