How would I prove that the sum and product of analytic functions are analytic?

With an analytic function being defined as...

A function is (real) analytic on an open setDin the real line if for any inDone can write...

in which the coefficients , , ... are real numbers and the series is convergent tofforxin a neighborhood of .

Thinking something along the lines of grouping the x terms together, then can find epsilon so that it converges after the nth term..? This along the right lines?