Existence of Interval of Convergence of Power Series/Corollary 2

Theorem
Let $\ds \map S x = \sum_{n \mathop = 0}^\infty a_n x^n$ be a power series.

Let $\map S x$ be convergent at $x = x_0$.

Then $\map S x$ is convergent for all $x$ such that $\sequence x < \sequence {x_0}$.

Proof
Let $\map S x$ converge when $x = x_0$.

Then by Existence of Interval of Convergence of Power Series, $x_0$ is in the interval of convergence of $\map S x$ whose midpoint is $0$.

The result follows by definition of interval of convergence.