Thread: Limit superior and limit inferior

1. Limit superior and limit inferior

I get the definition of limit superior and limit inferior, but I can't start a proof for these properties:

$\underline{\lim }{\,{x}_{n}}=-\overline{\lim }\left(-{{x}_{n}}\right)$

$\underline{\lim }{\,{x}_{n}}\le\overline{\lim }{\,{x}_{n}}$

$\underline{\lim }{\,{x}_{n}}+\underline{\lim }{\,{y}_{n}}\le\underline{\lim }\left({{x}_{n}}+{{y}_{n}}\right)\le\overline{\lim }{\,{x}_{n}}+\underline{\lim }{\,{y}_{n}}\le\overline{\lim }\left({{x}_{n}}+{{y}_{n}}\right)\le\overline{\lim }{\,{x}_{n}}+\overline{\lim }{\,{y}_{n}}.$

Hope you may enlight me.

2. Did you try to use the properties of $\inf$ and $\sup$ ?

3. Oh, not really.

How would you prove the third one? Can you show me?

Thanks!

4. Use $\displaystyle\inf_{n\geq k}(a_n+b_n)\geq \inf_{n\geq k}a_n+\inf_{n\geq k}b_n$ for the first inequality. For the second, you have to show that $\liminf (x_n+y_n)-\limsup x_n\leq \liminf y_n$. But $\liminf (x_n+y_n)-\limsup x_n =\liminf (x_n+y_n)+\liminf (-x_n)\leq \liminf (x_n+y_n-x_n)$.

5. So for the first one I use $\sup (-x_n)=-\inf x_n,$ by taking limits I get the result. Is it correct?

The second one follows easily from $\inf x_n\le \sup x_n,$ right?

6. You have to tell where you take the $\sup$. The definition is $\displaystyle \limsup_n x_n = \inf_{k\in\mathbb N}\sup_{n\geq k} x_n$.

7. Oh well, so I need to add that to what I wrote.

But by adding this, is it correct? If not, how to do it?