 Limit inferior

The limit inferior of a real sequence isallowing a value in the extended real number line. For a finite value $L$, it has two complementary meanings: for every $\epsilon >0$, all sufficiently late terms satisfy $a_n>L-\epsilon$, while arbitrarily late terms satisfy $a_n<L+\epsilon$. The first prevents persistent dips below $L$; the second prevents any larger number from being the true lower edge.

For $a_n=(-1{)}^n(1+1/n)$, the lower bounds in limit inferior and limit superior approach $-1$. Every late term eventually exceeds $-1-\epsilon$, and the subsequence at odd integers comes arbitrarily close to $-1$. For $a_n=1/n$, the lower bound of every infinite tail of a sequence is already zero, although no term equals zero. Thus a limit inferior need not be the minimum of any tail.