Comparison Test

Theorem
Let $\displaystyle \sum_{n=1}^\infty b_n$ be a convergent series of positive real numbers.

Let $\left \langle {a_n} \right \rangle$ be a sequence in $\R$.

Let $\forall n \in \N^*: \left|{a_n}\right| \le b_n$.

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

Corollary
Let $\displaystyle \sum_{n=1}^\infty b_n$ be a convergent series of positive real numbers.

Let $\left \langle {a_n} \right \rangle$ be a sequence in $\R$.

Let $H \in \R$.

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

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

Proof
Let $\epsilon > 0$.

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

So $\displaystyle \exists N: \forall n > N: \sum_{k = n+1}^\infty b_k < \epsilon$.

Let $\left \langle s_n \right \rangle$ be the sequence of partial sums of $\displaystyle \sum_{n=1}^\infty a_n$.

Then $\forall n > m > N$:

So $\left \langle s_n \right \rangle$ is a Cauchy sequence and the result follows from Convergent Sequence is Cauchy Sequence.

Proof of Corollary
Let $\epsilon > 0$.

Then $\frac \epsilon H > 0$.

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

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

Let $\left \langle s_n \right \rangle$ be the sequence of partial sums of $\displaystyle \sum_{n=1}^\infty a_n$.

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

So $\left \langle s_n \right \rangle$ is a Cauchy sequence and the result follows from Convergent Sequence is Cauchy Sequence.