Let us define limit superior of a bounded sequence $(x_n)$ in the following $2$ ways:
$1.$ $\limsup(x_n)=\lim\limits_{n\to \infty}\sup\{x_k:k\geq n\}$.
$2.$ It is the unique number $x\in \mathbb R$ such that for each $\epsilon>0,\exists n_0\in \mathbb N$ such that $x_n<x+\epsilon$ for all $n\geq n_0$.
I want to show that the two definitions are equivalent.Now,$(1)\implies(2)$ is easy .But how to show that the property $2$ is satissfied by a unique real number and $(2)\implies (1)$.I am not sure how to proceed.
Addendum
I do not think that the property $(2)$ is satisfied by unique number,any number greater than supremum satisfies this property.