Proving S2n-Sn is greater than a half in terms of the harmonic series

[gottfried]

Homework Statement

Let S_n = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}...+\frac{1}{n}. Show that |S_2n-S_n|\geq \frac{1}{2}

Homework Equations

The Attempt at a Solution

So I'm going to try and use induction

Base case let n=1

|S_2n-S_n| = \frac{1}{2}

So true for base case

Assume true for case that n=m

So if n=m+1

|S_2m+2-S_m+1|= |S_2m+2-S_2m+S_2m-S_m+1|

Our induction assumption tells us that |S_2m-S_2m+2|\geq \frac{1}{2} and S_2m+2-S_2m = \frac{1}{2m+1}+\frac{1}{2m+2}> 0

Therefore |S_2m+2-S_m|= |S_2m+2-S_2m+S_2m-S_2m+2|> \frac{1}{2}

Have I used induction properly? Since I haven't used the base case at all the proof feels a little like I've made the assumption that it is true and then shown that it is true but at the same time I feel like I haven't broken any of the rules of induction. I'm I wrong?

 

[LCKurtz]

Homework Statement

Let S_n = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}...+\frac{1}{n}. Show that |S_2n-S_n|\geq \frac{1}{2}

Homework Equations

The Attempt at a Solution

So I'm going to try and use induction

Base case let n=1

|S_2n-S_n| = \frac{1}{2}

So true for base case

Assume true for case that n=m

So if n=m+1

|S_2m+2-S_m+1|= |S_2m+2-S_2m+S_2m-S_m+1|

Our induction assumption tells us that |S_2m-S_2m+2|\geq \frac{1}{2} and S_2m+2-S_2m = \frac{1}{2m+1}+\frac{1}{2m+2}> 0

I don't quite follow that last step. But since $\{S_n\}$ is an increasing sequence you have$$
|(S_{2m+2}-S_{2m}) + (S_{2m}-S_{m+1})|=(S_{2m+2}-S_{2m}) + (S_{2m}-S_{m+1})\ge \frac 1 2$$since the quantities in parentheses are positive.
[Edit]: Ignore the highlighted paragraph. I wasn't completely awake. Look below:

But this problem hardly needs induction. Write out $S_{2n}-S_n$, look at how many terms there are, and underestimate each by the smallest one.

 

[gottfried]

So there are n terms between S_2n and S_n the smallest of which is \frac{1}{2n}

It follows that |S_2n - S_n|>n(\frac{1}{2n} ) =\frac{1}{2}

That is frustratingly simple. I feel silly but thanks a lot.