I know that if $\limsup_{x\to\infty}f(x)<L$ then there exist $x_{0}$ such that $f(x)<L$ for all $x\geq x_{o}$.
Is the following true:
If $\limsup_{x\to\infty}f(x)<L$ then, for any $\epsilon>0$, there exist $x_{0}$ such that $f(x)<L+\epsilon$ for all $x\geq x_{o}$.
(If the above is true, does $x_{o}$ depends on $\epsilon$?) Thanks