I have homework regarding finding the infimum and supremum of a set, but the examples from class and examples I can find online all describe the elements of the set as being bounded between values. What are some tips for finding the infimum and supremum when bounds are not given explicitly? For example, this is my homework problem:
Find the infimum and supremum of A={ $\frac{m+2n}{2m+n}$ : m,n $\in$ $\mathbb{N}$]. Since m and n are natural numbers, I know they have to be positive whole numbers (greater than zero), but I can't figure out where to go from there in order to find the inf and sup.
Any advice helps! Thank you :)
Play with the expression, to write it in a simpler form. For example, divide top and bottom by by $n$, to obtain the equivalent expression $$ \frac{(\frac mn) +2}{2(\frac mn) +1}$$ This tells you that the expression is a function of the ratio $m/n$. Now you don't need to worry about varying $m$ and $n$ separately.
If you now write $$f(x):=\frac{x+2}{2x+1}$$ then your set $A$ is the set of all $f(x)$ where $x$ is positive and rational.
Since this is a question about inf and sup for set $A$, you are looking for the global min and max for $f(x)$ as $x$ varies over positive rational numbers. This should remind you of using calculus.
The twist here is calculus will optimize $f(x)$ over all real $x>0$. You can still use calculus to determine the largest and smallest possible values for $f(x)$ for real $x>0$, but you need to show that these largest/smallest values can be also approached through rational $x$.