Show that a Sequence is monotonically decreasing

[Calu]

Homework Statement

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

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

 

[Ray Vickson]

Homework Statement

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

I believe I have to show a_n ≥ a_n+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 $\sqrt{n+1}$?

 

[Calu]

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 a_n≥a_n+1 as required, however I'm not sure how to write this in a more coherent way.

 

[pasmith]

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 a_n≥a_n+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 ...