Proof of triangle inequality: (a+b)^2 = a^2+2ab+b^2 < or = |a|^2 + 2|a||b|+|b|^2 = (|a| + |b|)^2 Taking the square root of both sides and remember that |x|=square root of (x^2), we can prove that |a+b| < or = |a| + |b| (Triangle inequality) ================================= This is the proof given in my text book, but the terrible thing is that I don't get the first line of the proof after reading it over many times. Q1) Why can you say that a^2+2ab+b^2 < or = |a|^2 + 2|a||b|+|b|^2 is true? Q2) I know that a^2 + 2ab + b^2 = (a+b)^2 with the absolute value signs. But is it true that |a|^2 + 2|a||b|+|b|^2 = (|a| + |b|)^2 ? Will the absolute values make any difference? I hope someone can explain them to me! Any help is appreciated.