Question

hello

how can i prove that |x+y|<=|x|+|y|

thanks in advance

Answer 1

Here , we have to prove |x+y| = |x|+|y|

we know that anything in mod comes out with positive value .

we have 4 assumptions .

1. x & y are negative

2. x & y are positive

3. x is positive y is negative

4. x is negative y is positive .

so in case one .

| -x + (-y) | = x+y

and |-x| + |-y| = x+ y

in case 2

|x+y| = x+y

and |x| + |y| = x+y

in case 3

|x + (-y)| = x+y

and |x| + |-y| = x+y

in case 4

| (-x)+y| = x+y

and |(-x)| + |y| = x+y .

so , in all cases , we found that

|x+y| = |x|+|y|

proved .