Proof by Contradiction: Arithmetic Mean & Geometric Mean

[shairozsohail]

Great! You are amazingly skilled at giving clear and concise explanations! 

[warbyvlog]

Thank your for the clarity, awesome job!

[usmanalam]

This is a great way of tackling this problem. It makes more sense and is layed out in a more progressive manner than the more direct approach I decided to take. In itself this proof is quite basic when considering just two values or even a countable set of values. I wonder, however, if someone were to set out a proof for the AM-GM mean for n number of values, where would they begin? Starts to get messy when thinking of proof by induction and gets even messier when taking a direct approach with logarithms. :o

Thanks for the video, by the way. Having another way to think of a problem really makes you see the depth of a problem and understand it better. Great job!

[macdondb]

Great video and proof, thank you. What if you start by assuming [sqrt(a)-sqrt(b)]^2>=0 (strictly greater if a != b)? Then you just square the LHS and solve in terms of a+b/2 to get the same result.

[sushruttrivedi]

gr8 video Thnx ... it would help in trignometry

[LOLxUnique]

I can see why some students cringe when they see this type of proof poop up in an exam. It really requires a different way of thinking when compared to the calculus that is heavily emphasized in high school maths. My peers showed me a proof for this exact problem, they instead assumed an identity and used it to prove the statement. To what extent are we allowed to use this method of proof in extension 1 mathematics to prove inequality statements?

[AeanHD]

Isn't there a possibility that both statements (> and <) are wrong?

[aashsyed]

is he teaching a class or just making a video?