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

With an analytic function being defined as...

A function $\displaystyle f$ is (real) analytic on an open setDin the real line if for any $\displaystyle x_0$ inDone can write...

$\displaystyle f(x) = \sum_{n=0}^\infty a_n \left( x-x_0 \right)^n = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + a_3 (x-x_0)^3 + \cdots$

in which the coefficients $\displaystyle a_0$, $\displaystyle a_1$, ... are real numbers and the series is convergent tofforxin a neighborhood of $\displaystyle x_0$.

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?