= Limit inferior
{title2=$\liminf$}
The limit inferior of a real <sequence> is
$$
\liminf_{n\to\infty}a_n=\sup_N\inf_{n\geq N}a_n,
$$
allowing a value in the <extended real number line>. For a finite value $L$, it has two complementary meanings: for every $\varepsilon>0$, all sufficiently late terms satisfy $a_n>L-\varepsilon$, while arbitrarily late terms satisfy $a_n<L+\varepsilon$. 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 $-1-\varepsilon$, 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.
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