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## Homework Statement

Suppose that ##( s_n )## and ## (t_n)## are bounded sequences. Given that ##A_k## is an upper bound for ##\{s_n : n \ge k \}## and ##B_k## is an upper bound for ##\{t_n : n \ge k \}## and that ##A_k + B_k## is an upper bound for ##\{s_n + t_n : n \ge k \}##, show that ##\sup \{s_n + t_n : n \ge k \} \le \sup \{s_n : n \ge k \} + \sup \{t_n : n \ge k \}##

## Homework Equations

## The Attempt at a Solution

Here is what I know. Since ##A_k##, ##B_k## and ##A_k + B_k## are upper bounds, we can conclude that ##\sup \{s_n + t_n : n \ge k \} \le A_k + B_k##, that ##\sup \{s_n : n \ge k \} \le A_k## and that ##\sup \{t_n : n \ge k \} \le B_k##. I think this might be a start, but I am not sure where to go from here...