I'm having trouble starting this proof from the definition I know that there exists a limit point $x$ such that $M \ge x > M- \frac \epsilon2$ but not quite sure how to go about it.
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For any $n$, we have $$\sup_{k\geqslant n}(a_k+b_k) \leqslant \sup_{k\geqslant n}a_k + \sup_{k\geqslant n}b_k,$$ as $$\{a_k+b_k : k\geqslant n\}\subset \{a_k+b_l:k,l\geqslant n\}$$ and $A\subset B$ implies $\sup A\leqslant\sup B$. Hence $$\begin{align*}\limsup_{n\to\infty} (a_n+b_n) &= \lim_{n\to\infty}\sup_{k\geqslant n}(a_k+b_k)\\&\leqslant \lim_{n\to\infty}\sup_{k\geqslant n}a_k+\lim_{n\to\infty}\sup_{k\geqslant n}b_k\\ &= \limsup_{n\to\infty}a_n + \limsup_{n\to\infty} b_n.\end{align*}$$
Math1000
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If they represent $\gamma, \alpha$ and $\beta$ correspondingly and if $\gamma \gt \alpha+\beta$, then there should be m such that $\alpha+\beta\lt a_m +b_m \le\gamma$ which means $(\alpha-a_m)+(\beta-b_m)\lt 0$.
Hamid Enki
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