# 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.

Ray Vickson
Homework Helper
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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
Homework Helper
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 ...