The book I'm reading showed a the promise without proving it, after the bolanzo Weirestrass theorem. The theorem is : Let $\left\{{a_n}\right\}^\infty _{n=1}$ , $\left\{{b_n}\right\}^\infty _{n=1}$ be real number sequences Prove: $$\scriptsize\ \lim_{\overline{n\to\infty}}a_n+\lim_{\overline{n\to\infty}}b_n\le\lim_{\overline{n\to\infty}}(a_n+b_n)\le\lim_{\overline{n\to\infty}}a_n+\overline{\lim_{n\to\infty}}b_n\le\overline{\lim_{n\to\infty}}(a_n+b_n)\le\overline{\lim_{n\to\infty}}a_n+\overline{\lim_{n\to\infty}}b_n$$
The first third and fifth are obvious by signing $$\overline{\lim_{\overline{n\to\infty}}}a_n=\overline{\underline{A}} , \overline{\lim_{\overline{n\to\infty}}}b_n=\overline{\underline{B}} $$ and get and solve it easily, what I can't understand how to prove is whats between those, my attempt at proving that $$\overline{\lim_\overline{n\to\infty}}(a_n+b_n)=\overline{\lim_\overline{n\to\infty}}a_n+\overline{\lim_\overline{n\to\infty}}b_n$$ by bolanzo Weiresterass we get that there exist sub sequences $a_{n_k},b_{n_k},a_{n_i},b_{n_i}$ such that $$\lim_{n\to\infty}a_{n_k}=\overline{A},\lim_{n\to\infty}a_{n_i}=\underline{A},\lim_{n\to\infty}b_{n_k}=\overline{B},\lim_{n\to\infty}b_{n_i}=\underline{B}$$ thus, $$\overline{\lim_{\overline{n\to\infty}}}(a_n+b_n)= \lim_{n\to\infty}(a_{n_{k,i}}+b_{n_{k,i}})=\lim_{n\to\infty}a_{n_{k,i}}+\lim_{n\to\infty}b_{n_{k,i}}=\underline{\overline{A}}+\overline{\underline{B}}$$ but then why would hthey write ≤ and not =?
EXtra: how do I prove $$\overline{\lim_\underline{n\to\infty}}(-a_n)=-\overline{\lim_\overline{n\to\infty}}a_n$$
