Question about an old result of Erdős and Simonovits My question concerns the  following result of Erd&#337;s and Simonovits.

A graph $G$ is $d$-almost-regular if $d\delta(G)\geq \Delta(G)$. Theorem 1 of the above paper states that a large $n$-vertex graph $G$ with $n^{1+\alpha}$ edges contains a large $d$-almost-regular subgraph $H$ for a constant $d$ depending only on $\alpha$. I cannot follow the proof of the theorem:

In case a) of the proof, if $C_1$ represents less than $\frac{1}{2}n^{1+\alpha}$ many edges, then we consider $G-C_1$. The resulting graph has only then at least $\frac{1}{2}n^{1+\alpha}$ edges, if $n$ refers to the order of the graph we are considering right now in the inductive step and not the order of the original graph (unless we are in the first step of the induction). 

When we later show that the induction terminates after a bounded number os steps, however, we estimate the number of edges by $\frac{1}{(4A)^k}n^{1+\alpha}$. This holds only true if $n$ refers to the order of the original graph throughout the induction. 

I tried to modify the proof such that case a) works, that is, $n$ refers to the order of the current graph in the inductive step. It is not hard to show that also then the induction terminates. However, with those modifications I cannot obtain that the subgraph $H$ is still large. 

Am I misunderstanding something? Is there a newer presentation of this theorem?
Yep. The writing is not great (but this is a nice result [but there should be an easier proof by now]). Anyway, you should read the $n^{1+\alpha}$ in the proof as a fixed quantity $\tau$, which is the number of edges in the original graph. The rest then follows.