# How come the proof is wrong if I do it backwards?

How come the proof is "wrong" if I do it backwards?

## Homework Statement

Let x >0. Then show that

$$x + 1/x \geq 2$$

and that the equality holds when x is 1

I got full marks on this, but remarked by my TA that I should do it backwards next time

Proof

$$x + 1/x \geq 2 \iff x^2 + 1 \geq 2x \iff x^2 - 2x + 1 \geq 0 \iff (x - 1)^2 \geq 0$$

Also the equality is true if x = 1

$$(1 - 1)^2 = 0^2 \geq 0$$

Q.E.D

My TA said I should start with $$(x - 1)^2 \geq 0$$ and go backwards. Why? If this was on an exam, how could I make up so much space and then erase and go back??

Sorry I meant the title should be

How come the proof is "correct" if I do it backwards?

## Homework Statement

Let x >0. Then show that

$$x + 1/x \geq 2$$

and that the equality holds when x is 1

I got full marks on this, but remarked by my TA that I should do it backwards next time

Proof

$$x + 1/x \geq 2 \iff x^2 + 1 \geq 2x \iff x^2 - 2x + 1 \geq 0 \iff (x - 1)^2 \geq 0$$

Also the equality is true if x = 1

$$(1 - 1)^2 = 0^2 \geq 0$$

Q.E.D

My TA said I should start with $$(x - 1)^2 \geq 0$$ and go backwards. Why? If this was on an exam, how could I make up so much space and then erase and go back??

You won't need to write it down and then erase it and go back. You'll simply use the symbol $\iff$ to indicate that you can do it backwards. If all the steps you do in a proof are also valid in the reversed order then you can do the reasoning backwards. Can't you? the main point is that all those statements are "if and only if" statements, so they are logically equivalent.

Since you used iff arrows, I don't see anything wrong with your proof. However, I think your TA just wanted you to start by writing something that you know is true (i.e. (x+1)2 ≥ 0) and then work towards what you're trying to prove. The way you did it makes it look like you started by assuming what you were trying to prove.

But like I said, the statements are iff, so I don't really see a problem.

The way you did it makes it look like you started by assuming what you were trying to prove.

What does that mean?

micromass
Staff Emeritus
Homework Helper

If this was on an exam, how could I make up so much space and then erase and go back??

Write a draft first on a separate peace of paper. Once you did that, you know how much space you need and you can make a clean copy.

What does that mean?

You should never start a proof by assuming that what you're trying to prove is true. So you shouldn't do this:

Claim: When x > 0, x+1/x ≥ 2

Proof: Let x>0. Assume x+1/x ≥ 2 ...

If your TA wasn't paying attention and missed that you used if and only if statements, he might have thought that you were assuming the claim was true and going from there.

Write a draft first on a separate peace of paper. Once you did that, you know how much space you need and you can make a clean copy.

A pain, but a necessity!

I like Serena
Homework Helper

You did it exactly right and the way I would recommend to do it.

micromass
Staff Emeritus
Homework Helper

You did it exactly right and the way I would recommend to do it.

However, I always tell my students not to use $\Leftrightarrow$. It's too dangerous. It's easy to say two things are equivalent when they are really not. I always ask my students to construct two proofs: one in the forward and one in the backward direction. This eliminates a lot of mistakes. It's a personal preference though and I would certainly not mark it incorrect!!

HallsofIvy
Homework Helper

You certainly can use "synthetic proof" (Starting with what you want to prove and ending with an obviously true statement) as long as you are careful that every step is "reversible". As long as it is given that x> 0, there is nothing wrong with
$$x+ \frac{1}{x}\ge 2$$
$$x^2+ 1\ge 2x$$
$$x^2- 2x+ 1\ge 0$$
$$(x- 1)^2\ge 0$$.

I like Serena
Homework Helper

I like to start in a case like this with what you want to proof, and use single arrows to a true statement.

Afterward, convert each single arrow to a double arrow, while checking if it is legal to do so.

This way, it is clear to a reader (and yourself) why and how the proof starts, and he can verify for himself if the reverse implications hold.