Ordering of Series of Ordered Sequences/Proof 1

Proof
Let $\sequence {\epsilon_n}$ be the real sequence defined by:
 * $\forall n \in \N : b_n - a_n$

From Linear Combination of Convergent Series, $\ds \sum_{n \mathop = 0}^\infty \epsilon_n$ is convergent with sum $\epsilon > 0$.

Then:

Hence the result, by definition of inequality.