Limit sup limit inf

Question

I am getting confused about the definition of $\limsup$. It's been given in the books that if $M$ is the $\limsup x_n$ then $x_n < M+\varepsilon$ except finitely many $n$ i.e. for all $n > N$ and $x_n> M-\varepsilon$ for infinitely many $n$. I am having problem in understanding the difference between “except finitely many $n$” & “infinitely many $n$”.

Answer 1

Consider , for instance, the sequence $(x_n)_{n\in\mathbb N}=\left(\dfrac1n+(-1)^n\right)_{n\in\mathbb N}$. Its $\limsup$ is $1$. Now, take $\varepsilon=\dfrac12$. Then:

• the inequality $x_n<1+\varepsilon$ takes place except for finitely many $n$'s; actually, it fails only when $n=2$;
• the inequality $x_n>1-\varepsilon$ takes place for infinitely many $n$'s: it occurs when (and only when) $n$ is even.