Both statements are consequences of the Cauchy Integral Formula. For the first, let be continuous on a contour and define
for each not on Then is analytic at each point not on the contour, and the derivitive is
See if you can prove this from Cauchy. Then do some reasoning to conclude that all the derivatives exist for an analytic function.
The statement of your second question is called Liouville's Theorem, and it is a beautiful theorem of analysis. To prove it, one generally proceeds by letting be analytic inside and on a circle of radius centered at If has a maximum of on this circle, then
for This lemma is proved by first using Cauchy's Formula on , then noticing that the integrand of the resulting integral is bounded by Then use the standard bound
where is your contour of length From this, you should arrive at the equation given by the lemma. Liouville's Theorem follows by taking and letting go to infinity. A simple bit of reasoning leads to the conclusion that is constant.
Post again if you get stuck on any details.