14. proof of the triangle inequality. (a) verify that the triangle inequality is true for several different real numbers x and y. be sure to have some examples where the real numbers are negative. (b) explain why the following proposition is true: for each real number r, r jrj. (c) now let x and y be real numbers. apply the result in part (14b) to both x and y. then add the corresponding parts of the two inequalities to obtain another inequality. use this to prove that jx c yj jxj c jyj