A sparse graph has a prescribed-size induced subgraph of bounded maximum degree

If $m=1$, then any single vertex set $X$ works, because a one-vertex graph has maximum degree $0=4ϵ(m-1)$.

If $ϵ=0$, then $G$ has no edges, so any $m$-vertex set $X$ has maximum degree $0=4ϵ(m-1)$. Hence we may assume from now on that $m\ge 2$ and $ϵ>0$. Average the edge count over all $(2m-1)$-vertex subsets $Y\subseteq V(G)$. Some such $Y$ satisfies $e(G[Y])\le ϵ(2m-1)(m-1)$, because the expected edge count in a random $(2m-1)$-subset is exactly the global edge count multiplied by the probability that both endpoints of a given edge are chosen.

In this chosen set $Y$, fewer than $m$ vertices have degree greater than $4ϵ(m-1)$. Otherwise at least $m$ vertices would contribute more than $4ϵ(m-1)$ each to the degree sum, giving $2e(G[Y])>4ϵm(m-1)>2ϵ(2m-1)(m-1)$, contrary to step 1.2.

Delete all vertices of $Y$ whose degree in $G[Y]$ exceeds $4ϵ(m-1)$. By step 2.1 at least $m$ vertices remain; choose any $m$ of them and call the resulting set $X$. Every vertex of $G[X]$ has degree at most its degree in $G[Y]$, so $\mathrm{\Delta }(G[X])\le 4ϵ(m-1)$.

This $X$ has the required size and degree bound.