I'm stuck at a proof about Graphs in the context of Graph Spanners. A Lemma says:
Let $G$ be a graph with $n$ vertices and $m$ edges. If $G$ has girth more than $2k$, then $m\leq n^{1+\frac{1}{k}}$.
The proof is made by contradiction where repeatedly any node from $G$ of degree at most $\lceil n^\frac{1}{k}\rceil$ and any edges incident to that node is removed, until there is no such node anymore. This step leads to a subgraph $G'$ of $G$ of minimum degree more than $\lceil n^\frac{1}{k}\rceil$.
The conclusion is that $G'$ must have girth at most $2k$ and therefore also $G$, which is a contradiction.
I don't understand the conclusion: Why must $G'$ have girth at most $2k$? Why is it not possible that $G$ has girth more than $2k$ and minimum degree more than $\lceil n^\frac{1}{k}\rceil$?
I don't really see an answer to this question, so I would be very grateful if someone can help me.