need help w/ proofs: Let x, y, and z be real numbers. Then

[luckyc1423]

1) Let x, y, and z be real numbers. Prove the following:

a) if x>1, then x^2>x
b) if 0<x<1, then x^2<1

Here is what I have done....I know its not correct could someone help me...

a) x(x) > x, so x>1

b) x(x) < 1

x < 1/x for every 0<x<1

2) Prove that, if x >=0 and x<= E for all E > 0, then x = 0

No clue on (2).

 

[mark07]

1a) You have the idea but when you write proofs, be careful about the logical statements. Proving "If x>1, then x^2>x" means that you assume x>1 and try to prove that x^2>x.

Then, like you said, (x)(x) > (1)(x), gives the desired statement. What I mean is that when you finish up by saying "so x>1" at the end, it sounds like you were trying to prove x>1!

1b) Even though it is obvious, you haven't quite finished the proof. These are obvious statements but the reason you are asked to prove them is to get you used to the logic behind proofs and how to write them accurately.

Assume 0<x<1. Then

(x)(x) < (1)(1)

x^2 < 1.

 

[mark07]

2) Let \(\displaystyle \L x\geq 0\) and \(\displaystyle \L x \leq \varepsilon\) for all \(\displaystyle \L \varepsilon >0\).

\(\displaystyle \L x\geq 0\) means \(\displaystyle \L x= 0\) or \(\displaystyle \L x> 0\).

If \(\displaystyle \L x= 0\), we are done.

If \(\displaystyle \L x> 0\), choosing \(\displaystyle \L \varepsilon = \frac{x}{2}\) gives a contradiction (why?? write explanation), therefore this case is not possible.