Proof help: $\inf_m a_m + \inf_n b_n \leq \inf_n (a_n+b_n)$ and $\inf_m a_m + \inf_n b_n = \inf _{m,n} a_m + b_n$

Question

so I've had a lot of trouble with this problem, specifically on the notation side. I'll list my questions below the given question which is:

Let $ A = \{a_n \mid n\in \mathbb{N}\}$ and $B= \{b_n \mid n\in \mathbb{N}\}$ be two bounded sets in $ \mathbb{R} $. Show that

$\inf_m a_m + \inf_n b_n \leq \inf_n (a_n+b_n)$ $~~~~~~~~~~$(1)

$\inf_m a_m + \inf_n b_n = \inf _{m,n} a_m + b_n$.$~~~~~~~~$(2)

So for the most part it makes a lot of sense to me that $\inf(A+B)\leq\inf(A)+ \inf(B)$ for any set $A$ and $B$.

Because of this I more or less understand how I would prove (1) however it the notation that includes $m,n$ is a bit confusing and then I have absolutely no idea how and why $a_m$ on the LHS of (1) becomes $a_n$ on the RHS.

I'm hoping once I understand (1), I'll understand how to go about (2) but any hints would be great.

In class we have yet to go over limits and it's quite strange I cannot find anything looking like this problem online that doesn't utilize limits to solve/prove this. But I'd appreciate if that can stay the case. Sorry about the super basic question, but this has stumped me for days. Thank you!

Edit: I'll add my proof(s) below :) with some simple examples that helped me understand the notation and question.

Answer 1

To assert your confusion with the indices your first intuition is absolutely right. You could completely rewrite your first equation as follows: $$ \inf_m a_m +\inf_n b_n\le \inf_p(a_p+b_p), $$ with different indices for each $\inf$. In most cases, when setting a quantifier over integers people like to use the letter $n$ and when having two such quantifiers, we often use $m$ and $n$. So the labelling is purely a question of preference but has no true meaning in what you want to prove.

A way to prove your first equation would be noting that for any $p\in \mathbb N$ we have: $$ \inf_m a_m +\inf_n b_n \le a_p+b_p. $$ This holding for any $p$, you have in particular that $$ \inf_m a_m +\inf_n b_n\le \inf_p(a_p+b_p) $$ which is exactly your first equation.

If you need any further details, hit me up!

Answer 2

I believe I got somewhere for the proof of (2)

Proof of (2) Let $k=\inf_n{a_n}$ such that $k\leq a$ for all $a\in A$ and

Let $l=\inf_m{b_m}$ such that $l\leq b$ for all $b\in B$.

We want to show that $k+l=\inf_{m,n}{a_m+b_n}$ so that $k+l\leq a+b$ for all $a\in A$ and $b\in B$. and that there does not exist $r+q\in \mathbb{R}$ such that $k+l<r+q\leq a+b$ for all $a\in A$ and $b\in B$.

By construction $k+l$ is a lower bound as $k\leq a$ for all $a\in A$ and $l\leq b$ for all $b\in B$ making it true that $k+l\leq a+b$ for all $a\in A$ and $b\in B$.

Now we will show that $k+l$ is the greatest lower bound, infimum, by showing there does not exist $k+l<r+q\leq a+b$ for all $a\in A$ and $b\in B$.

Suppose there exists $r+q\in \mathbb{R}$ such that $k+l<r+q\leq a+b$ for all $a\in A$ and $b\in B$.

Since $k$ is the least upper bound of $A$, there does not exist $r\in \mathbb{R}$ such that $k<r\leq a$ for all $a\in A$ and

since $l$ is the least upper bound of $B$, there does not exist $q\in\mathbb{R}$ such that $l<q\leq b$ for all $b\in B$.

Therefore there does not exist $r+q\in\mathbb{R}$ such that $k+l<r+q\leq a+b$ for all $a\in A$ and $b\in B$ thus $k+l\leq a+b$ for all $a\in A$ and $b\in B$ making $k+l$ the infimum of $A+B$ and thus $k+l=\inf_{m,n}{a_m+b_n}$.

End of Proof.