# Show that a Sequence is monotonically decreasing

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## Homework Statement

I was wondering how I would go about showing that (an) is monotone decreasing given that an = 1/√n.

I believe I have to show an ≥ an+1, but I'm not sure how to go about doing that.

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Ray Vickson
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## Homework Statement

I was wondering how I would go about showing that (an) is monotone decreasing given that an = 1/√n.

I believe I have to show an ≥ an+1, but I'm not sure how to go about doing that.
Well, PF rules require you to make a start on your own. What do you know about the magnitudes of $\sqrt{n}$ and $\sqrt{n+1}$?

Well, PF rules require you to make a start on your own. What do you know about the magnitudes of $\sqrt{n}$ and $\sqrt{n+1}$?
I know that the magnitude of $\sqrt{n+1}$ is larger than that of $\sqrt{n}$. Therefore I would assume that the opposite would be true for the magnitude of their reciprocals which would make an≥an+1 as required, however I'm not sure how to write this in a more coherent way.

pasmith
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I know that the magnitude of $\sqrt{n+1}$ is larger than that of $\sqrt{n}$. Therefore I would assume that the opposite would be true for the magnitude of their reciprocals which would make an≥an+1 as required, however I'm not sure how to write this in a more coherent way.
(1) $\sqrt{n + 1} > \sqrt{n}$.
(2) If $n > 0$ then dividing both sides by $\sqrt{n} > 0$ preserves the inequality. Hence $\frac{\sqrt{n + 1} }{\sqrt{n}} > 1$.
(3) Dividing both sides by ... > 0 preserves the inequality. Hence ...