Can the Mean Value Theorem be Used to Prove Inequalities for Natural Numbers?

[Johnny Blade]

Homework Statement

I need to prove the following for all natural numbers:

1/(x+1) < ln(x+1) - ln(x) < 1/x

Homework Equations

The Attempt at a Solution

It's in the part of the mean value theorem problems, I try using it didn't go any where. I tried thinking of other ways, but nothing seems to work.

 

[HallsofIvy]

The first thing I would do is write ln(x+1)- ln(x) as ln((x+1)/x). Are you saying that x must be a positive integer? Then I would assume that you should use proof by induction, not the mean value theorem!

 

[lurflurf]

Induction would work nicely, but taking advantage of your section hint try
log(x+1)-log(x)=[log(x+1)-log(x)]/[(x+1)-x]=log'(x+t)=1/(x+t)
where t is some number such that 0<t<1
so the problem is reduced to showing that
when ever 0<t<1
1/(x+1)<1/(x+t)<1/(x+0)

 

[Johnny Blade]

I got it with the mean value theorem, but because that's for all real numbers, can I say it's true for natural numbers too because it is a subset of the real numbers? Thanks for the help.