Increasing and decreasing prime gaps

Let $p_n$ denote the $n$th prime and $g_n:=p_{n+1}-p_n$ the $n$th prime gap. We demonstrate the existence of infinitely many values of $n$ for which $g_n>g_{n+1}>\cdots >g_{n+m}$ with $m\gg \log \log \log n$ and similarly for the reversed inequalities. In doing so we settle a conjecture of Erd"os for the case $m=2$.