Limit inferior

A limit inferior 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 infimum and supremum on the tails of the sequence. Together with the limit superior , it describes the full range of subsequential behavior via subsequences .

Examples:

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