In the definition for limit superior of real functions on this page we have:
Let $f : (a, \infty) \to \mathbb{R}$. The limit superior as $x \to \infty$ is defined as $\displaystyle{\limsup_{x \to \infty} f(x) = \lim_{x \to \infty} \sup_{t \geq x} \{ f(t) \} = \inf_{x \geq a} \left \{ \sup_{t \geq x} \{ f(t) \} \right \}}$
I am wondering if the last equality is wrong, namely $\displaystyle{\inf_{x \geq a} \left \{ \sup_{t \geq x} \{ f(t) \} \right \}}$, if the function is defined on an interval $(a, \infty)$ shouldn't it be $\displaystyle{\inf_{x > a}}$ instead of $\displaystyle{\inf_{x \geq a}}$?