Comparison Test

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

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

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

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

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

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

Let $$H \in \mathbb{R}$$.

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

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

Proof
Let $$\epsilon > 0$$.

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

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

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

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

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

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

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.