Question

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...\[\]

Answer 1

i guess the question mark above should be \(x^2+2|x||y|+y^2\), but what does this tell me!?\[\]

Answer 2

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??\[\]

Answer 3

Your reasoning seems sound to me.

Answer 4

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

Answer 5

thank you!!!

Answer 6

yw