Limit superior

A limit superior of a real sequence $(a_n)_{n\ge 1}$ is the extended real number

provided the right-hand side is interpreted in $[-\infty,\infty]$.

This definition is built from repeated use of supremum and infimum on the “tails” of the sequence. It packages subsequential behavior: values near $\limsup a_n$ are realized along suitable subsequences .

Examples:

• If $a_n=(-1)^n$, then $\limsup_{n\to\infty} a_n=1$.
• If $a_n=\tfrac1n$, then $\limsup_{n\to\infty} a_n=0$.