Properties of infimum and supremum of monotone bounded sequences

Question

Suppose $x = (x_1, x_2, ...)$ is a bounded increasing sequence and $x \in l^\infty$. Thus its limit is the least-upper bound of the sequence, i.e. $sup(x_i), i \in \Bbb N $.

Is it always true for such a sequence that $|sup(x_i)|\le sup(|x_i|) $?

Answer 1

First Suppose that $\sup a_i \geq 0$. Let $|a_i|$ be the sequence of absolute values. Then we must have $$ a_i \leq |a_i|.$$ Hence $$ 0\leq \sup a_i \leq \sup |a_i|$$ and if you take absolute value in both sides you have the result.

Second Suppose $\sup a_i \leq 0$. Then $$ \sup a_i \leq |a_i|, \text{for every}~i $$ and so $$ \sup a_i \leq \sup |a_i| $$ and by taking the absolute value in both sides you have the result.