- #1

- 56

- 0

## Homework Statement

prove

[tex] |arctan(x)-arctan(y)| \leq |x-y| [/tex]

## The Attempt at a Solution

I tried separating cases:

1. both x and y are in the same quadrant;

2. x and y are not in the same quadrant;

3. x and y are zero (trivial);

4. either x or y is zero.

Now, i can see why each of these is true from drawing arctan (the asymptotic behavior and |arctan(x)| =< |x| being important), but i'm having trouble producing a mathematical argument.. I tried doing something with the derivative but i just get confused. Also, proving the equality |arctan(x)| =< |x| is the _next_ question, so i doubt i'm supposed to use it here.