Comparison Test/Corollary 1

Corollary to Comparison Test
Let $\ds \sum_{n \mathop = 1}^\infty b_n$ be a convergent series of positive real numbers.

Let $\sequence {a_n}$ be a sequence in $\R$ or a sequence in $\C$.

Let $H \in \R$.

Let $\exists M: \forall n > M: \size {a_n} \le H b_n$.

Then the series $\ds \sum_{n \mathop = 1}^\infty a_n$ converges.

Proof
Let $\epsilon > 0$.

Then $\dfrac \epsilon H > 0$.

As $\ds \sum_{n \mathop = 1}^\infty b_n$ converges, its tail tends to zero.

So:
 * $\ds \exists N: \forall n > N: \sum_{k \mathop = n+1}^\infty b_k < \frac \epsilon H$

Let $\sequence {s_n}$ be the sequence of partial sums of $\ds \sum_{n \mathop = 1}^\infty a_n$.

Then $\forall n > m > \max \set {M, N}$:

So $\sequence {s_n}$ is a Cauchy sequence and the result follows from:
 * Real Number Line is Complete Metric Space

or:
 * Complex Plane is Complete Metric Space.