Suppose that $\displaystyle a_n\rightarrow L$. Show that if $\displaystyle a_n\leq M$ for all $\displaystyle n$, then $\displaystyle L\leq M$.

Prove by contradiction and by choosing $\displaystyle \epsilon =\frac{L-M}{2}$.

I have a standard proof for how to solve this, but not a contradictory proof (which is what I need). Can someone help with that?