Register to reply

Verification of simple inequality proof

by GeoMike
Tags: inequality, proof, simple, verification
Share this thread:
GeoMike
#1
Jul31-06, 09:43 PM
P: 70
The question given is:
If a < b, prove that a < (a+b)/2 < b

The book had a different proof than the one I came up with. I understand the book's proof, I just want to know if my proof is also ok.

I did the following:

a < (a+b)/2 < b
2a < a+b < 2b
a < b < 2b-a
a-b < 0 < b-a

Since it was given that a < b, a-b must be less than 0, and b-a must be greater than zero, so the inequality a < (a+b)/2 < b is true if a < b.

Is this ok?

Thanks,
-GeoMike-
Phys.Org News Partner Science news on Phys.org
'Smart material' chin strap harvests energy from chewing
King Richard III died painfully on battlefield
Capturing ancient Maya sites from both a rat's and a 'bat's eye view'
d_leet
#2
Jul31-06, 09:49 PM
P: 1,075
It looks a bit like you did the proof in reverse. It seems that you started with a<(a+b)/2<b and then got to a<b rather than starting with a<b and proving a<(a+b)/2<b like the problem seems to want.
Data
#3
Jul31-06, 10:21 PM
P: 998
really you should show the direction of the implications you're using, so what you mean is actually

[tex]a<(a+b)/2<b \Longleftarrow 2a<(a+b)<2b \Longleftarrow a<b<2b-a \Longleftarrow a-b<0<b-a \Longleftarrow a<b,[/tex]

and not the other way. It doesn't matter much here since all the inequalities there are equivalent (so the implications work both ways, ie. in fact [itex]a<b \Longleftrightarrow a<(a+b)/2<b[/itex]).

Office_Shredder
#4
Jul31-06, 11:16 PM
Emeritus
Sci Advisor
PF Gold
P: 4,500
Verification of simple inequality proof

Proof by reverse is great, but you have to show that the steps can logically be reversed at the end
HallsofIvy
#5
Aug1-06, 05:29 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,682
That's sometimes called "synthetic proof". It's often used to prove trig identitities. Start with what you want to prove and work back to an obviously true statement. It's valid as long as every stepe is reversible,.
GeoMike
#6
Aug1-06, 12:22 PM
P: 70
Thank you for the replies!

-GeoMike-


Register to reply

Related Discussions
Attempt at proof by contradiction need verification Precalculus Mathematics Homework 6
Another inequality proof.. Precalculus Mathematics Homework 4
Proof this inequality using Chebyshev's sum inequality Calculus & Beyond Homework 1
Inequality proof help Calculus & Beyond Homework 0
Inequality proof General Discussion 7