Proving Inequalities: Tips & Tricks from Dave

[daviddoria]

I never have much luck when something says "prove something <= something else". I usually just fiddle around and occasionally get lucky and reduce it to a constant < an express I know can't be less than that constant. But most times I can't reduce it to something like that. Is there any kind of "principled" approach that someone can suggest/ point me to read about?

As an example, say it was something like
ln(1+a) \leq a^2 b

How would you approach something like this?

Thanks,

Dave

 

[csprof2000]

You can, in general, use any number of arguments to show something like that.

For the natural log one, just raise e to both sides. Then you get

1 + a = exp(b*a^2)

Then you can write

1 = exp(b*a^2) - a.

I don't know... then just pull out an exponential term and get

1 = exp(b*a^2)(1 - a*exp(-b*a^2)) and argue that, as either A or b grow without bound, the right hand simplifies to exp(b*a^2) and the left remains constant. And for a, b both positive, this is clearly true.

I mean, it's just algebra, some logic, and knowing what you're trying to show. Does that help?