Theorem: (SRL)
For every $\epsilon>0$ and integer $m\geq 1$ there is an $M$ such that every graph $G$, with $|G|\geq m$ has an $\epsilon$-regular partition $V(G)=V\_0\cup\ldots\cup V\_k$ for some $m\leq k\leq M$. 


Can someone explain to me why this statement is not trivial? For instance, what stops me choosing $M$ larger than $|G|$ and picking $k=|G|$, so I can split $G$ up into singletons, which is trivally $\epsilon$-regular for any $\epsilon>0$. 



Szemeredi's Regularity Lemma
Quantifier error. You have to fix your M before you are given a graph G; whereas your approach would require one to have the graph G at hand, before choosing M.


