Proving "Bounds of log(n)" Inequality

[Deano10]

Homework Statement

Prove that for any integer n >= 2,

1/2 + 1/3 + ... + 1/n <= log(n) <= 1 + 1/2 + 1/3 + ... + 1/(n-1)

Homework Equations

None

The Attempt at a Solution

I can see pictorally why the inequality holds true but despite numerous am struggling to make any real progress! Any hints or tips on how to get started would be very much appreciated!

 

[futurebird]

Maybe look at a series expansion for the log:

http://www.efunda.com/math/exp_log/series_exp.cfm

 

[Hurkyl]

I can see pictorally why the inequality holds true

What is the pictoral reason?

 

[Deano10]

The picture I had in mind was of that of log (n) with step functions (of the values in the inequalities) both above and below the graph drawn out by log (n).

It is this that leads me to think the proof must involve the use of bounding step functions, but I cannot see how to begin.

 

[Deano10]

I have had a further look at this and think that I can adapt the proof of the fact that the limiting difference between the harmonic series and natural logarithm tending to the Euler constant to prove one side of the inequality.

However, this still leaves the other side of the inequality unsolved and the fact that I am not sure this is the approach I should be taking!

Any hints would be most appreciated...