Proving the Cauchy-Schwarz Inequality

[VertexOperator]

Homework Statement

Prove that:

Homework Equations

\sum_{k=1}^{n}x_{k}^{2}\geq \frac{1}{n}\left ( \sum_{k=1}^{n}x_{k} \right )^{2}

The Attempt at a Solution

I am not sure what to do to be honest. But it looks like the Cauchy–Schwarz inequality to me.

 

[micromass]

Oh come on. I'm sure there are things you can try. For example, solve the case n=2. Or try induction. Or something.

 

[VertexOperator]

Oh come on. I'm sure there are things you can try. For example, solve the case n=2. Or try induction. Or something.

Sigma is intimidating :(
But I will try doing it.

 

[micromass]

Sigma is intimidating :(
But I will try doing it.

Yes, I agree that it is intimidating. That is why I usually do some special cases first like n=2 and n=3 to get a feeling for the proof!

 

[VertexOperator]

I got 2x_{1}^{2}+2x_{2}^{2}\geq x_{1}^{2}+2x_{1}x_{2}+x_{2}^{2}

 

[micromass]

I got 2x_{1}^{2}+2x_{2}^{2}\geq x_{1}^{2}+2x_{1}x_{2}+x_{2}^{2}

Do you see why this is true?

 

[VertexOperator]

Do you see why this is true?

Is it because \left ( x_{1}^{2}+x_{2}^{2} \right )\geq 2x_{1}x_{2}
But why is this true?
PS: Is it because \left ( x_{1}-x_{2} \right )^{2}\geq 0 \therefore \left ( x_{1}^{2}+x_{2}^{2} \right )\geq 2x_{1}x_{2}

 

[VertexOperator]

Do I just do normal induction from here?

 

[micromass]

After thinking a little bit about it, you won't need induction at all.

Basically, you need to prove that

\sum_{k=1}^n x_k \leq \sqrt{n}\sqrt{\sum_{k=1}^n x_k^2}

You were completely right that this resembled Cauchy-Schwarz. So, try to apply Cauchy-Schwarz somewhere.

Of course, for that inequality, you need two sequences. The sequence (x_1,...,x_n) is one of them, what is the other?

 

[VertexOperator]

After thinking a little bit about it, you won't need induction at all.

Basically, you need to prove that

\sum_{k=1}^n x_k \leq \sqrt{n}\sqrt{\sum_{k=1}^n x_k^2}

You were completely right that this resembled Cauchy-Schwarz. So, try to apply Cauchy-Schwarz somewhere.

Of course, for that inequality, you need two sequences. The sequence (x_1,...,x_n) is one of them, what is the other?

Is it x_{1}^{2},...,x_{n}^{2} ?

And thank you a lot for the help. I didn't think about squarerooting both sides :( I really need to learn to recognized this stuff.

 

[micromass]

No, it's not that.

Think of a very easy (constant) sequence.

 

[VertexOperator]

No, it's not that.

Think of a very easy (constant) sequence.

Although this question can be done using the CS Inequality, it can be done this way too:
a^2+b^2 \geq 2ab

[We need to sub into a and b, all the combinations of 2 numbers from the set \ \ x_1, \ x_2, \ x_2 \ \dots, \ x_n

x_1^2+x_1^2 \geq 2x_1 x_2

x_1^2+x_3^2 \geq 2x_1x_3

.
.
.

x_1^2+x_n^2 \geq 2x_1x_n

x_2^2+x_3^2 \geq 2x_2 x_3

And so on, then we add these inequalities side by side.

(n-1)(x_1^2+x_2^2 \dots + x_n^2) \geq 2(x_1x_2+x_1x_3+ \dots)

n(x_1^2+x_2^2+\dots+x_n^2) \geq x_1^2+x_2^2+\dots + x_n^2 + 2(x_1x_2+\dots

The RHS is the expanded form of:

n\sum_{k=1}^{n} x_k^2 \geq \left(\sum_{k=1}^{n}x_k \right)^2

\sum_{k=1}^{n} x_k^2 \geq \frac{1}{n}\left(\sum_{k=1}^{n}x_k \right)^2

 

[VertexOperator]

Could you please show me how you would do it using the CS inequality?

 

[micromass]

You need to show that\sum_{k=1}^n x_k \leq \sqrt{n}\sqrt{\sum_{k=1}^n x_k^2}

The CS inequality gives something like

\sum_{k=1}^n x_ky_k \leq ...

If I use the same x_k in both inequalities above, then what does the corresponding y_k need to be?

 

[VertexOperator]

You need to show that


\sum_{k=1}^n x_k \leq \sqrt{n}\sqrt{\sum_{k=1}^n x_k^2}

The CS inequality gives something like

\sum_{k=1}^n x_ky_k \leq ...

If I use the same x_k in both inequalities above, then what does the corresponding y_k need to be?

Ok, thank you. I will try to do it now.