Difference between lim and lim sup?

Question

I know it is a basic question but the definitions in POMA are too rigorous.

I need some sort of example to understand what's going on.

Can someone please help?

Thank you so much

Answer 1

Given a sequence $(a_n )_n$, a number $\ell$ is called a partial limit of $(a_n)_n$ if there exists a subsequence $\left(a_{n_k}\right)_k$ which converges to $\ell$. It can be shown that the set $$S=\{ \ell: \ell \text{ is a partial limit of }(a_n) \}$$ is closed, which furnishes the existence of $\max S$ and $\min S$. These are known as the limes superior and the limes inferior of $(a_n)$ respectively, or $\limsup a_n,\liminf a_n$. Note that the two always exist!

Now, in the special case that $\liminf a_n=\limsup a_n$ we have that all partial limits of $(a_n)$ (elements of $S$ defined above) coincide to a single number. In this case we say that $(a_n)$ itself has a limit, which is defined to be this common value.

EXAMPLE: The sequence $(a_n)=(-1)^n$ has partial limits $\pm 1$ obtained by the subsequences of odd/even indices. We thus have $$S=\{-1,+1\} $$ and $$\liminf a_n=-1 \\ \limsup a_n=+1. $$ In this case the sequence $(a_n)$ itself has no limit, also called divergent.