Suppose that is of class . Given and a positive number , show that there exists a polynomial such that:

For all .

The hint given was to choose a polynomial that approximates , and suppose that throughout the interval . Now say that , and to bound and in terms of both and .

So I'm thinking to choose that and try to find an antiderivative, and then find an antiderivative of that such that the other two conditions hold, but I'm having trouble seeing how those distances all relate to one another.