# Supremum/Infimum with an inequality subscript

I've been going through my notes on Limit Superior/Inferior, and all I've been able to find (both online and in books) is the definition $$\liminf_{n\longrightarrow\infty}a_n:=\lim_{n\longrightarrow\infty}\left(\inf_{k\geq n}a_k\right)$$ and a similar one for $\limsup$. But I have never come across the notation on the right-hand side, i.e. the inequality beneath the infimum. What does this denote? If I were to guess, I'd say it's $$\inf_{k\geq n}a_k=\inf\left\{a_n:n\leq k\right\},$$ i.e. the infimum of the first $k$ terms of $\left(a_n\right)_{n\in\mathbb N}$. But nowhere can I find this explicitly defined.

• Your guess is right, but, in the curly brackets, it should say "$a_k$". It's not explicitly defined because the notation is self-explanatory. – MathematicsStudent1122 Nov 6 '16 at 8:32
• @MathematicsStudent1122, But what is $k$? Don't we usually take a sequence $\left(a_n\right)_{n\in\mathbb N}$ such that $n$ takes on any value in $\mathbb N$? Wouldn't the condition $n\leq k$ on $a_k$ therefore be vacuous? – Luke Collins Nov 6 '16 at 8:38

## 1 Answer

No. We have $$\inf_{k\geq n}a_k=\inf\left\{a_k:n\leq k\right\} =\inf \{a_n, a_{n+1},....\}$$