Show $\limsup_{n\rightarrow \infty }(-a_{n}) = -\liminf_{n\rightarrow \infty }(a_{n})$

Question

Show $\limsup_{n\rightarrow \infty }(-a_{n}) = -\liminf_{n\rightarrow \infty }(a_{n})$ for a sequence $(a_{n})\subset\mathbb{R}$

Attempt: $\limsup_{n\rightarrow \infty }(-a_{n}) = \inf_{n\geq1 }(\sup\begin{Bmatrix} -a_{n},-(a_{n+1}),... \end{Bmatrix} )$

Now using the fact $-\sup(-A) = \inf(A)$ where $A\subset\mathbb{R}$

$= \inf_{n\geq1 }(-\inf\begin{Bmatrix} a_{n},a_{n+1},... \end{Bmatrix} )$ $= -\inf_{n\geq1 }(\inf\begin{Bmatrix} a_{n},a_{n+1},... \end{Bmatrix} )$

I would like to show $-\inf_{n\geq1 }(\inf\begin{Bmatrix} a_{n},a_{n+1},... \end{Bmatrix} )= -\sup_{n\geq1 }(\inf\begin{Bmatrix} a_{n},a_{n+1},... \end{Bmatrix} ) = -\liminf_{n\rightarrow \infty }(a_{n})$ but im not sure the previous inf(-inf) = -inf(inf) step is even correct. Still trying to wrap my head around this concept!

Answer 1

We can rewrite $\limsup$ as \begin{align*} \limsup_{n \to \infty} (-a_n) &= \lim_{n \to \infty} \sup_{k > n} (-a_k) \\ &\stackrel{(*)}{=} \lim_{n \to \infty} -\inf_{k > n} (a_k) \\ &= -\lim_{n \to \infty} \inf_{k > n} (a_k) \\ &= -\liminf_{n \to \infty} (a_n) \\ \end{align*} where $(*)$ follows from the fact that $\sup (-A) = -\inf (A)$, as you've stated.