Give an example to show that the inequalities are strict inequalities

Give an example to show that the following three inequalities $$\liminf_{n \to \infty} (a_n) +\liminf_{n \to \infty} (b_n)\le\liminf_{n \to \infty} (a_n+b_n)\le\limsup_{n \to \infty} (a_n+b_n) \le \limsup_{n \to \infty} (a_n) + \limsup_{n \to \infty}(b_n)$$ are all strict inequalities.

Can anyone please explain how I answer this question? I do not understand which example to give to satisfy the above or how I should go about it. Any help would be appreciated thank you.

Solution 1:

I'll get you started. Let $a_n=(-1)^n$ and $b_n=(-1)^{n+1}$. Then $\liminf_{n\rightarrow\infty}(a_n)=\liminf_{n\rightarrow\infty}(b_n)=-1$. But $\liminf_{n\rightarrow\infty}(a_n+b_n)=0$.

Note that you don't need the same sequences to satisfy all inequalities.