Limit superior of a sequence

Let \((x_n)_{n\geq 1}\) be a sequence of real numbers. Its limit superior is \[ \limsup_{n\to\infty} x_{n} \coloneqq \lim_{N \to \infty} \sup_{n>N} x_{n} \in [-\infty,\infty], \] where the limit exists in \([-\infty,\infty]\) because \(\sup_{n>N}x_n\) is nonincreasing in \(N\).