Prove $\limsup\limits_{n \to \infty} (a_n+b_n) \le \limsup\limits_{n \to \infty} a_n + \limsup\limits_{n \to \infty} b_n$

Define for all natural numbers $k$: $A_k = \sup\{ a_n: n \ge k \}$, $B_k = \sup\{ b_n: n \ge k \}$ (where $A_k, B_k \in \mathbb{R} \cup \{+\infty\}$, (They are decreasing because for larger $k$ we take the $\sup$ of fewer terms), so that by definition $\limsup_{n \to \infty} a_n = \lim_{k \to \infty} A_k$ and similarly for $B_k$ and $\limsup_{n \to \infty} b_n$. Also we consider the $C_k = \sup \{ (a_n + b_n) : n \ge k \}$, so that $\lim_{k \to \infty} C_k = \limsup_{n \to \infty} (a_n+b_n)$.

Now, fix an index $k$, then for all $n \ge k$ we have $a_n + b_n \le A_k + B_k$, because we estimate $a_n$ by the supremum of all terms of $(a_n)$ with $n \ge k$ and likewise for the $b_n$. As (for fixed $k$) the right hand side is fixed:

$$C_k = \sup \{ (a_n + b_n : n \ge k \} \le A_k + B_k\mbox{.}$$

This holds for all $k$, so we take the $\inf$ or $\lim$ on both sides as $k$ tends to infinity, and this preserves the inequality and we are done.

I happened to making solutions for PMA, so I would like to share my solution here:

If either $\limsup_{n \to \infty} a_n = +\infty$ or $\limsup_{n \to \infty} b_n = +\infty$, there is nothing to prove. So we may assume $\limsup_{n \to \infty} a_n = A, \limsup_{n \to \infty} b_n = B$, where $A < +\infty, B < +\infty$ (but each of them can possibly take $-\infty$).

Given $\varepsilon > 0$, there exist $N_1, N_2 \in \mathbb{N}$, such that $a_n < A + \varepsilon/2$ for all $n \geq N_1$ and $b_n < B + \varepsilon/2$ for all $n \geq N_2$. Take $N = \max(N_1, N_2)$, it follows that for all $n \geq N$, $$a_n + b_n < A + \varepsilon/2 + B + \varepsilon/ 2 = A + B + \varepsilon.$$ Let $n \to \infty$ in the above equation, we conclude that $\limsup_{n \to \infty} (a_n + b_n) \leq A + B + \varepsilon$. Since $\varepsilon$ is arbitrary, it follows that $\limsup_{n \to \infty}(a_n + b_n) \leq A + B$, proving the result.

Hint:

Given two sequences, $\;\displaystyle \{a_n\}_{n \in \Bbb N},\;$ $\,\{b_n\}_{n \in \Bbb N},\;$

and given the definition of the supremum of a sequence, we can see that for every $k\geq n$, $$(a_k + b_k) \;\; \leq \;\;\sup_{k\geq n} a_k + \sup_{k\geq n} b_k\,.$$

Now how does this imply that $$\lim_{n\to \infty} \sup(a_n + b_n) \;\; \leq \;\; \lim _{n \to \infty} \sup a_n + \lim_{n\to \infty} \sup b_n\quad ?$$

Added: see Definition 3.16, Theorem 3.17, 3.19: perhaps you'd prefer to use the notation used there.