[The Comparison Test for Series (2.7.4)] Assume $a_n$ and $b_n$ are sequences satisfying $0 \leq a_n \leq b_n$ for all $n \in \mathbf{N}$.

If $\sum_{n=1}^{\infty} b_n$ converges, then $\sum_{n=1}^{\infty} a_n$ converges.
If $\sum_{n=1}^{\infty} a_n$ diverges, then $\sum_{n=1}^{\infty} b_n$ diverges.

For the absolute value function definition and other properties see here.
For the definitions of sequences and what it means to for a sequence to converge, see this, for subsequences see this.
For the definitions of series, and what it means to for a series to converge, see this.
For the “show the limit” template and an example, see this.

Formal Proof

Let $(a_n)$ and $(b_n)$ be sequences such that $0 \leq a_n \leq b_n$ for all $n \in \mathbf{N}$. For (i), we are given that $\sum_{n=1}^{\infty} b_n$ converges. Therefore, by the Cauchy criterion for series, we know that there exists a number $N \in \mathbf{N}$ such that

$$ \begin{align*} |b_{m+1} + b_{m+2} + ... + b_n| < \epsilon \quad \text{whenever $n, m \geq N$}. \end{align*} $$

but $b_n \geq 0$ by assumption so we can drop the absolute value from the term. Moreover, we know that each term $b_n$ is greater than $a_n$ for all $n \in \mathbf{N}$ and so

$$ \begin{align*} |b_{m+1} + b_{m+2} + ... + b_n| &< \epsilon \\ a_{m+1} + a_{m+2} + ... + a_n \leq b_{m+1} + b_{m+2} + ... + b_n &< \epsilon \\ \end{align*} $$

From this we can see that $|a_{m+1} + a_{m+2} + ... + a_n| < \epsilon$ which means that the series $\sum_{n=1}^{\infty} a_n$ must converge by the Cauchy criterion for series. For (ii), note that it is the contrapositive of (i). $\blacksquare$

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