Limit superior and limit inferior of a sequence

Let \((a_n)_{n\geq 0}\) be a sequence of real numbers. Its limit superior and limit inferior are \[ \begin{aligned} \limsup_{n\to\infty} a_n &\coloneqq \lim_{n\to \infty} \sup_{j\geq n} a_j = \inf_{n\geq 0} \sup_{j\geq n} a_j, \\ \liminf_{n\to\infty} a_n &\coloneqq \lim_{n\to \infty} \inf_{j\geq n} a_j = \sup_{n\geq 0} \inf_{j\geq n} a_j, \end{aligned} \] with values in \([-\infty,\infty]\); the limits exist and equal the infimum and supremum because \(\sup_{j\geq n}a_j\) is nonincreasing and \(\inf_{j\geq n}a_j\) is nondecreasing in \(n\).