Proving an Inequality: x+y>=2sqrt(xy)

[toslowtogofast2a]

Homework Statement

If x and y are both real numbers and both greater than 0 prove that x+y>=2sqrt(xy)

Relevant Equations

See attached image

See the attached image for my attempt. My main concern is can I assume that y > x prove it for that case and then show it is equal if y = x.

My whole proof is centered around y > x so if i cannot make that assumption then I have to start over. Let me know your thoughts. Thanks in advance for the help.

 

[anuttarasammyak]

Adding "Asssume that x >y all the above said stands by exchanging x and y. So we know the given relation stands." would be fine.

 

[toslowtogofast2a]

Adding "Asssume that x >y all the above said stands by exchanging x and y. So we know the given relation stands." would be fine.

thanks.

 

[pbuk]

Because the expression is symmetrical in $x$ and $y$ we would usually phrase the second line of your proof as "if $x \ne y$ assume without loss of generality $x > y$". See https://en.wikipedia.org/wiki/Without_loss_of_generality.

You don't then need to add anything else.

 

[WWGD]

Homework Statement: If x and y are both real numbers and both greater than 0 prove that x+y>=2sqrt(xy)
Relevant Equations: See attached image

See the attached image for my attempt. My main concern is can I assume that y > x prove it for that case and then show it is equal if y = x.

My whole proof is centered around y > x so if i cannot make that assumption then I have to start over. Let me know your thoughts. Thanks in advance for the help. View attachment 353406

You may want to look into the AGI : Arithmetic-Geometric Inequality:
https://en.m.wikipedia.org/wiki/AM–GM_inequality

 

[toslowtogofast2a]

Because the expression is symmetrical in $x$ and $y$ we would usually phrase the second line of your proof as "if $x \ne y$ assume without loss of generality $x > y$". See https://en.wikipedia.org/wiki/Without_loss_of_generality.

You don't then need to add anything else.

thanks for the link. that makes sense.

 

[toslowtogofast2a]

You may want to look into the AGI : Arithmetic-Geometric Inequality:
https://en.m.wikipedia.org/wiki/AM–GM_inequality

Thanks for the reply the proof in that link is the exact proof that was in my book. Since mine was different I started to question if I went wrong somewhere.