Long cycles in random subgraphs of graphs with large minimum degree

Let $G$ be any graph of minimum degree at least $k$, and let $G_p$ be the random subgraph of $G$ obtained by keeping each edge independently with probability $p$. Recently, Krivelevich, Lee and Sudakov showed that if $pk\to\infty$ then with probability tending to 1 $G_p$ contains a cycle of length at least $(1-o(1))k$. We give a much shorter proof of this result, also based on depth-first search.