We use induction on $n$. The case $n=1$ is immediate.

Every student attended session $n$, so each attendance total lies between $1$ and $n$. The $n$ totals are distinct, hence they are exactly $1,2,\ldots,n$.

The student with total $n$ attended every session and was the only student present at session $1$. Remove this student and session $1$, and renumber sessions $2,\ldots,n$ as $1,\ldots,n-1$.

The remaining sessions have exactly $1,2,\ldots,n-1$ attendees. No remaining student attended the deleted session, so their totals are unchanged and still pairwise distinct.

By the induction hypothesis, each remaining student attended every session from their first attendance onward. The removed student attended every session, so the conclusion holds for all students.