Proof in Set Theory: Subset of ]0,2]

[mtayab1994]

Homework Statement

{1/x+1/y / (x,y) in (IN*)^2} subset of ]0,2]

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.

 

[Mark44]

Homework Statement

{1/x+1/y / (x,y) in (IN*)^2} subset of ]0,2]

What is this supposed to mean?

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]

Do you mean to say that

\{1/x+1/y~\vert~x,y\in \mathbb{N}\setminus\{0\}\}

That would make sense...

So you need to show that

\frac{1}{x}+\frac{1}{y}\leq 2

for all naturals x and y. Maybe use the fact that

\frac{1}{x+1}\leq \frac{1}{x}

and do induction??

 

[mtayab1994]

Do you mean to say that

\{1/x+1/y~\vert~x,y\in \mathbb{N}\setminus\{0\}\}

That would make sense...

So you need to show that

\frac{1}{x}+\frac{1}{y}\leq 2

for all naturals x and y. Maybe use the fact that

\frac{1}{x+1}\leq \frac{1}{x}

and do induction??

Yep that's exactly what I wanted to say. thank you.