Proving a^2 or b^2 is Less than or Equal to n

[saadsarfraz]

Suppose n=ab, show that a^2<=n or b^2<=n.

 

[CRGreathouse]

Start with trichotomy: a < b, a = b, or a > b. It's pretty simple.

 

[saadsarfraz]

I'm fairly new to proofs, so can you please check if this is correct.

if a=b, n=a^2 or b^2=n
if a>b, a^2>ab, a^2>n
if b>a, b^2>ab, b^2>n

 

[CRGreathouse]

What you write is correct, though for the last two you should really use the opposite variable (and reverse signs) because you're trying to show that something's smaller, not bigger.

 

[mXSCNT]

Try multiplying a > b by b instead of by a.

 

[saadsarfraz]

so it should be like this then:

if a>b, ba>b^2, n>b^2
if b>a, ab>a^2, n>a^2