I do not understand proving some limit superior problem.

Question

Prob. Show that $~~~\displaystyle\limsup_{k\to\infty} (a_k+b_k) \le \limsup_{k\to\infty} a_k + \limsup_{k\to\infty} b_k$.

1. Let $A_j=\displaystyle\sup_{k\ge j}a_k,~~~B_j=\sup_{k\ge j}b_k,~~\text{ and} ~~~~ C_j=\sup_{k\ge j}(a_k+b_k)$.
2. Then, for $k\ge j, ~~~a_k+b_k\le A_j+B_j,~~$ since $A_j\ge a_k$ and $B_j\ge b_k$.
3. Thus, $C_j \le A_j+B_j$.
4. $$\lim_{j\to\infty}C_j\le \lim_{j\to\infty}(A_j+B_j) = \lim_{j\to\infty}A_j+ \lim_{j\to\infty}B_j$$ $$\Leftrightarrow\lim_{j\to\infty}\sup_{k\ge j}(a_k+b_k)\le \lim_{j\to\infty}\sup_{k\ge j}a_k + \lim_{j\to\infty}\sup_{k\ge j}b_k$$$$\Leftrightarrow\limsup_{k\to\infty}(a_k+b_k)\le \limsup_{k\to\infty}a_k + \limsup_{k\to\infty}b_k$$

I do not understand the process number three.

I think $C_j \ge (a_k+b_k)$ like $A_j$ and $B_j$.

How can I get $C_j \le A_j+B_j$ from the following three facts? $$A_j+B_j\ge a_k+b_k \quad \quad \quad \quad C_j \ge (a_k+b_k)$$

Answer 1

Let $k\ge j$ be arbitrary; we know that $a_k+b_k\le A_j+B_j$. In other words, $A_j+B_j$ is an upper bound for the set $\{a_k+b_k:k\ge j\}$. By definition

$$C_j=\sup_{k\ge j}(a_k+b_k)$$

is the least upper bound of that set, so $C_j$ is less than every other upper bound for the set. In particular, $C_j\le A_j+B_j$.