Prove:
lim $\displaystyle a_{n}$ > M --> $\displaystyle a_{n}$ > M for n>>1 (large n).
Then we have:
Since $\displaystyle \lim_{n\rightarrow\infty} a_{n} =l$>M THAT implies that for all ε>0 and hence for ε= l-M>0 ,there exists an N such that :
for all n , $\displaystyle n\geq N$ then $\displaystyle |a_{n}-l|< l-M$
or $\displaystyle M-l< a_{n}-l< l-M$.
Thus $\displaystyle a_{n}> M$ for large n biger or equal to N