# What is the flaw with this proof?

1. Oct 9, 2008

### soandos

x==Sqrt[0+Sqrt[0+Sqrt[x+Sqrt[0+...]]]]
x==Sqrt[x]
x==1 and zero
1==Sqrt[0+Sqrt[0+Sqrt[x+Sqrt[0+...]]]]==0
1==0

2. Oct 9, 2008

### CRGreathouse

Your third line incorrectly assumes that any solution to the second line is a solution to the first line. This is not the case.

Consider
x = 7
x * 0 = 7 * 0
0 = 0, which holds for all x (including 2)
2 = 7

3. Oct 10, 2008

### soandos

I am not multiplying by zero in any step though, i am just saying that Sqrt[0+Sqrt[0+Sqrt[x+Sqrt[0+...]]]]==x
means that Sqrt[0+x]==x

4. Oct 10, 2008

### CRGreathouse

You are not multiplying by zero, correct. You are saying that line 1 implies line 2, which is a correct statement. You further claim (implicitly) that an x-value solving line 2 solves line 1, which is incorrect.

5. Oct 10, 2008

### soandos

why does it not follow?

6. Oct 10, 2008

### CRGreathouse

Why would it?

If someone is 52, then they're in their 50s. If someone is in their 50s, they might be 58. If someone is 58, they are eligible to join the 55+ club. But it doesn't follow that someone who is 52 might be able to join the 55+ club.

If x = -3 then x^2 = 9. If x^2 = 9 it follows that x is 3 or -3. But it does not follow that both 3 = -3 and -3 = -3.

7. Oct 10, 2008

### epkid08

Just because the zeros of a function equal zero when put through the function, doesn't mean that the zeros equal each other.

(x+1)(x-1)=0
x=1;x=-1
1≠-1

8. Oct 11, 2008

### soandos

thanks.
btw epkid08, that is not what i was saying. i was saying that value of the function was x, not the zeros of f(x)

9. Oct 11, 2008

### epkid08

It was an analogy. The message was that if x = {a,b}, a≠b.

x=√(x)
x={0,1}
0≠1

10. Oct 12, 2008

### CRGreathouse

My message was that if F(x) --> G(x), and G(x) holds precisely for x in X, F(x) need not hold for all x in X.