You've been way too generous with the inequalities. You have to choose your approximations wisely, or else they won't be bounded anymore. That's the hard part in this kind of question: proving an inequality in two variables. Proving inequalities isn't something you can just do by the book, it's different every time and usually done with wizardry and eldritch skills. There's absolutely no general guideline on how to do that, but here's some general tricks you should try:
- Use both sides of your inequality, don't stuff everything onto one side.
- Break the symmetry of symmetrical inequalities: If you have an inequality that is symmetrical in x and y, then you only need to consider the case that x >= y, the other case works identically.
- Occasionally it helps to substitute variables.
For example, let's have a look at the first function, and show that it's Lipschitz continuous with a factor of 0.5 (we could have chosen any larger number too but we're doing 0.5 for the thrill):
We need to show
|sqrt(x+1)-sqrt(y+1)| <= 0.5|(x-y)|
We can assume x>=y because of symmetry, and since the square root function is increasing, we need to show
2(sqrt(x+1)-sqrt(y+1)) <= x-y.
Let's substitute x' = sqrt(x+1), y' = sqrt(y+1). We get
2(x' - y') <= x'^2 - y'^2.
But hold up, we can factor the right side as (x'+y')(x'-y')! Since both sides are positive and x'-y' is positive too, we can divide by x' - y' and it remains to show that
2 <= x' + y'.
This is true, because x' >= 1 and y' >= 1. We're done!
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Topic: Mathematics Help Thread (Read 226899 times)