Advanced Calc/Analysis: Delta Epsilon proof

[stvoffutt]

Homework Statement

Using the definition of |x-a|<delta implies |f(x) - L|<epsilon, prove that lim x->0 x^n*sin(1/x) holds for all n belonging to natural numbers.

Homework Equations

Definition of a limit

The Attempt at a Solution

Ok, so when I see "prove for all n belonging to natural numbers" I immediately think induction. So this is what I have done so far. For n=1 lim x->0 x^n*sin(1/x) is true; the limit is zero. Now I will assume lim x->0 x^n*sin(1/x) for some n is true, then I need to show that n+1 is also true. So I start using the definition of the limit and don't know what my L should be and how to use induction along with this definition of a limit. Please help and thank you in advanced.

 

[JThompson]

$$\mbox{If you can prove that }\lim_{x\to 0}x^{n}sin\left(\frac{1}{x}\right)\mbox{ is bounded by two functions whose limit is L, then you've proven that the limit exists.}$$
$$\mbox{(You don't have to invoke the Squeeze Theorem.)}$$

$$\mbox{Start with }-1\leq \sin\left(\frac{1}{x}\right)\leq 1\mbox{. What can you say about }x^{n}sin\left(\frac{1}{x}\right)\mbox{ ?}$$

$$\mbox{It isn't completely necessary, but it might help you see what L to use if you make the substitution x=1/t, then }$$

$$\lim_{x\to 0}x^{n}sin\left(\frac{1}{x}\right) = \lim_{t\to\infty}\left(\frac{1}{t}\right)^{n}sin\left(t\right)$$