Studying sequences, lim sup and lim inf

Question

If we have a sequence $x_n$ such that $\lim \sup|x_n − l| = 0$, $l$ being a real number,

What can we say about it?

Is it the same if $\lim \inf|x_n − l| = 0 $

Answer 1

$ \liminf | x_n - l | = 0 $ does not imply anything about the convergent of the sequence. Consider a sequence $x_n$, where $x_n = |\sin \frac{\pi}{2}n|$. Then $x_n$ equals 1 or 0, depending on if $n$ is odd or even; the sequence does not convergence but has two limit points 0 and 1, making 0 it's lower limit.

$ \limsup | x_n - l | = 0 $ is equivalent to the convergence of the sequence, though. Note that the upper and lower limits of sequences always exist, and the fact that absolute value is definitely larger than zero, we have $0 \leq \liminf | x_n -l | \leq \lim|x_n-l| \leq \limsup |x_n -l| = 0 $, Which implies $ \lim|x_n-l|= 0$, that is, the sequence converges to l.

This is actually a useful technique to show convergence.