Ordering of Series of Ordered Sequences

Theorem
Let $\sequence {a_n}$ and $\sequence {b_n}$ be two real sequences.

Let $\ds \sum_{n \mathop = 1}^{\infty} a_n$ and $\ds \sum_{n \mathop = 1}^\infty b_n$ be convergent series.

For each $n \in \N$, let $a_n < b_n$.

Then:
 * $\ds \sum_{n \mathop = 0}^\infty a_n < \sum_{n \mathop = 0}^\infty b_n$