How Do You Prove This Inequality Using Induction?

[Boombaard]

Homework Statement

Show that $$2( \sqrt{n+1} - \sqrt{n} ) < \frac{1}{ \sqrt{n}} < 2( \sqrt{n} - \sqrt{n-1}), n \in N$$

2. The attempt at a solution

this works for n=1, but I'm entirely at a loss when trying to figure out which equations i can assume to be 'obviously' true (for the sake of the induction step), and what cannot be assumed..
All sides of the inequality eventually become 0, and while it seems fairly obvious to me that they all do it at an incrementally slower pace than the preceding one, I'm not sure what to make of this mathematically, or how to mold it into an equation expressing that.. can anyone lend a hand?

@radou: sorry, took a few mins to figure out what i'd done wrong

 

[radou]

You may want to correct your LaTeX code.

 

[AlephZero]

Unless the question says "prove by induction", a direct proof is easy.

Take each inequality separately, multiply both sides by \root n, and it should be fairly obvious what to do after that.

Another direct proof: think about approximations to
$$\int x^{-1/2} \, dx$$

 

[dextercioby]

I can't possibly see how induction could do it.

Daniel.

 

[tim_lou]

indeed, a direct proof is much much easier than induction... the inequalities are equivalent to:
$$\sqrt{n+1}-\sqrt{n}>0$$

 

[AlephZero]

I suppose you could argue that a direct proof IS a proof by induction.

You say "assume the result is true for n = k-1", then show it is true for n = k ... even though you never actually used the assumption

 

[Boombaard]

bah. i feel like I'm really not very bright here, but why is the difference between the sqrts of 2 successive natural numbers always < .5 ?

 

[matt grime]

multiply out:

$$(\sqrt{n+1}-\sqrt{n}) \frac{\sqrt{n+1}+\sqrt{n}}{\sqrt{n+1}+\sqrt{n}}$$