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So the following are infinite series of positive term, satisfying $0\le a_n\le b_n$ .$$\sum_{n=1}^{\infty} a_n \,, \, \sum_{n=1}^{\infty} b_n$$ Suppose that $\sum_{n=1}^{\infty}b_n$ is convergent. Can anyone help with concluding that $\sum_{n=1}^{\infty}a_n$ is also convergent.

I think i may have to use the squeeze principle, or perhaps the limits of the series (seeing as $b_n$ is convergent), but not sure how to go about this.

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    $\begingroup$ This is called the Direct Comparison Test $\endgroup$ – Hayden May 13 '14 at 13:42
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    $\begingroup$ Hint: sequence of partial sums of the first series form is increasing and bounded (by sum of the second) $\endgroup$ – Marcin Łoś May 13 '14 at 13:44
  • $\begingroup$ @Marcin_Łoś Cheers for that! Never really thought of partial sums of them. $\endgroup$ – Paulistic May 13 '14 at 13:49
  • $\begingroup$ @Hayden Just had a quick Google of that and its exactly what I was looking for. Don't know how I missed it flicking through notes. $\endgroup$ – Paulistic May 13 '14 at 13:50
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Let us prove something analogous

Let $a_n$ and $b_n$ be two sequences. $b_n$ and $a_n+b_n$ converges then $a_n$ converges

Proof: Note that: If $x_n$ and $y_n$ are two convergent sequences $x_n+y_n$ and $x_n-y_n$ converges. In the previous statement take $x_n=a_n+b_n$ and $y_n=b_n$ and infer that $x_n-y_n=a_n+b_n-b_n=a_n$ converges.

To prove the series version of the theorem all you have to do is to consider the partial sum of the series and apply the same proof there.

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