Cant understand this step in a bounded prove

[transgalactic]

the question:
f(x) continues on $$(-\infty,a]$$
and suppose that the border $$\lim_{x->-\infty}f(x)$$ exists and finite.
prove that f(x) is bounded on $$(-\infty,a]$$ and/or that exists
$$x_0\epsilon(-\infty,a]=\lim _{x->-\infty }f(x)$$
so
$$\sup_{x\epsilon(-\infty,a]} f(x)$$
in other words prove that f(x) gets its highest value on $$(-\infty,a]$$
and the supremum is the maximum

the non understood part:

suppose
$$\lim_{x->-\infty}f(x)=m_0$$
suppose $$m_0<a$$
and we check on the interval of $$[m_0,a]$$ where $$[m_0,a]\subseteq (-\infty,a]$$

they prove by a counter example that:
"suppose the function is not bounded from the top then $$\forall n\epsilon N$$ and
$$m_0\leq x_n\leq a$$"

i can't understand it.if a function is bounded by some epsilon then we take N for which after this N (n>N) f(x)<epsilon
if its not bounded from the top then
f(x) is bigger then epsilon for the whole interval

this is not what they writee up there
what are they writing there??

 

[Office_Shredder]

What's x_n, and why are you looking at all of them? I'm not sure I understand what's going on in your post

 

[transgalactic]

how its supposed to be diverged?