# Supremum and infimum of series

The question asks to show the equality/inequality of supremum/infimum of series.

Suppose that $$\{a_{n,m}\}$$ are nonnegative real numbers for all $$n,m \in \mathbb{N}$$.

1. Suppose that for each $$n$$, $$m \mapsto a_{n,m}$$ is a nondecreasing function of $$m$$, i.e., $$a_{n,m_1} \leq a_{n,m_2}$$ when $$m_1 \leq m_2$$. Show that $$\sup_{m \in \mathbb{N}} \sum_{n = 1}^\infty a_{n,m} = \sum_{n =1 }^\infty \sup_{m \in \mathbb{N}} a_{n,m}$$ Regardless of whether or not the sides are finite or infinite.

2. Suppose that for each $$n$$, $$m \mapsto a_{n,m}$$ is nonincreasing, i.e., $$a_{n,m_1} \leq a_{n,m_2}$$ for all $$n \in \mathbb{N}$$ and all $$m_2 \leq m_1$$. Show that $$\sum_{n =1 }^\infty \inf_{m \in \mathbb{N}} a_{n,m} \leq \inf_{m \in \mathbb{N}} \sum_{n = 1}^{\infty} a_{n,m}$$ Give an example to show that the inequality is strict.

My attempt in part 1 is to show $$\sup_{m \in \mathbb{N}} \sum_{n = 1}^\infty a_{n,m} \leq \sum_{n =1 }^\infty \sup_{m \in \mathbb{N}} a_{n,m}$$ and also $$\sum_{n =1 }^\infty \sup_{m \in \mathbb{N}} a_{n,m} \leq \sup_{m \in \mathbb{N}} \sum_{n = 1}^\infty a_{n,m}$$. I am not sure how to apply the concept of supremum/infimum to equality/inequality of series. Any hints would be very helpful.

For any $$n,m$$, $$a_{n,m} \leq \sup_r\,a_{n,r}$$. Therefore $$\sum_n{a_{n,m}} \leq \sum_n{\sup_r\,a_{n,r}}$$ and the rest is textbook definition of sup to get the first inequality.
For the reverse inequality, show that for every $$N \geq 1$$, $$\sum_{n=1}^N{\sup_r\,a_{n,r}} \leq \sup_r\,\sum_{n=1}^N{a_{n,r}}$$ (show that in this case the sups are limits), then that the LHS is not greater than $$\sup_r\,\sum_{n \geq 1}{a_{n,r}}$$, and conclude.
For the counter-example, find for instance an example where all $$a_{n,m}$$ converge to zero, but all the $$\sum_n{a_{n,m}}$$ are infinite.