## anonymous 4 years ago my professor asked me to prove the triangle inequality by taking into consideration that, and i quote, "$$(x+y)^2=x^2+2xy+y^2\leq|x|^2+2|x||y|+|y|^2=?$$." i have no clue what he is expecting...

1. anonymous

i guess the question mark above should be $$x^2+2|x||y|+y^2$$, but what does this tell me!?

2. anonymous

or perhaps $$(|x|+|y|)^2$$... hmm... now that i remember, he said something about $$|x|=\sqrt{x^2}$$... oh... then perhaps $$|x+y|=\sqrt{(x+y)^2}$$ and i already showed that $$(x+y)^2\leq(|x|+|y|)^2$$ which ends up becoming $$|x+y|\leq|x|+|y|$$. is this it??

3. asnaseer

Your reasoning seems sound to me.

4. asnaseer

there are some interesting alternative proofs as well. see here: http://math.ucsd.edu/~wgarner/math4c/derivations/other/triangleinequal.htm

5. anonymous

thank you!!!

6. asnaseer

yw