The discussion revolves around proving that the set {1/x + 1/y | (x,y) in (IN*)^2} is a subset of the interval ]0,2]. The subject area is set theory and inequalities involving natural numbers.
The discussion is active, with participants clarifying the notation and confirming understanding of the problem. There is a suggestion of using mathematical induction to approach the proof, but no consensus on a specific method has been reached yet.
Participants are working under the assumption that x and y are natural numbers greater than zero, and there is some uncertainty regarding how to formally prove the inequality.
What is this supposed to mean?mtayab1994 said:Homework Statement
{1/x+1/y / (x,y) in (IN*)^2} subset of ]0,2]
mtayab1994 said:Homework Equations
The Attempt at a Solution
When x=y=1 u get a sum of 2 which is in ]0,2] and for any x and y greater than 1 u get a sum between 0<sum≤2.
It's a simple problem but i just don't know how to show the proof. Some help please.
micromass said:Do you mean to say that
[tex]\{1/x+1/y~\vert~x,y\in \mathbb{N}\setminus\{0\}\}[/tex]
That would make sense...
So you need to show that
[tex]\frac{1}{x}+\frac{1}{y}\leq 2[/tex]
for all naturals x and y. Maybe use the fact that
[tex]\frac{1}{x+1}\leq \frac{1}{x}[/tex]
and do induction??