Take the tour ×  How can we prove that $\liminf x_n = -\limsup
(-x_n)$?  share|improve this question What definition of
$\liminf$ and $\limsup$ do you take? What did you try? –  julien
Mar 18 at 20:35 first you need to prove that Inf (-A)=-Sup (A)
–  cswinson Mar 18 at 20:37 lim Sup xn= lim (Sup {xn : n>N}) – 
cswinson Mar 18 at 20:41 add comment  1 Answer  I'll go with the
definition you mention. Defining $\liminf x_n$ as the smallest
limit of all converging subsequences in $[-\infty,+\infty)$ and
$\limsup$ as the greatest etc... would make the statement
slightly easier to check.  Assume you have proved that $-\sup
(-A)=\inf A$ for every subset $A$ of $\mathbb{R}$. Applying this
to the set $A=\{x_n\;;\;n>N\}$ yields: $$
-\sup_{n>N}(-x_n)=\inf_{n>N}x_n\qquad \forall N. $$ Taking the
limit on both sides gives the formula. Even in the case where
these sequences are constant equal to $-\infty$, that is
$\liminf x_n=-\infty$ and $\limsup (-x_n)=+\infty$.  Now let us
prove the set property. Clearly, $A$ is not bou