1. ## Proofs

a) prove | |x| - |y| | <= | x - y |
b) prove If | x-y | < c, then |x| < |y| + c
c) prove If | x-y | < e for all e>0, then x = y
d) X1....Xn are real numbers prove |X1 = X2 + ... + Xn| <= |X1| + |X2| + ..... + |Xn|

2. #1
|x|=|x-y+y|<=|x-y|+|y| -> |x|-|y|<=|x-y|
Likewise |y-x|<=|y|-|x| -> |x|-|y|>= -|y-x|= -|x-y|
Thus ||x|-|y||<=|x-y|.

#3
Suppose that x<>y. Then |x-y|>0 so let e=|x-y|.

3. I follow problem one of what you did really well. I am confused on 3

what is x<>y....does this just mean x = y, also why are you letting e = |x-y|, doesnt it say e is less than that....I know I am completely wrong because you have been right ever since you have started helping me. Maybe if you could explain it I would understand

4. Well if x is not y then |x-y| is a positive number. Right?
But for each positive number, e, |x-y|<e so |x-y|<|x-y|.
But that is impossible! So x must be y.

5. ok, ic that makes alittle more sense that you explained it.

X1....Xn are real numbers prove |X1 + X2 + ... + Xn| <= |X1| + |X2| + ..... + |Xn|

6. If you followed #1, then just apply the triangle inequality several times.

7. Originally Posted by luckyc1423
b) prove If | x-y | < c, then |x| < |y| + c
|x-y|<c

|x-y|+|y|<c+|y|

|x|=|x-y+y|<=|x-y|+|y|<c+|y|

Thus,
|x|<c+|y|

8. Should these questions be easier for me to answer? I mean you guys come up with the solutions really quickly and I spend hours without even figure out how to do it. But I am at the average as far as grades for the course so I guess im not the only one struggling in the class.

9. Originally Posted by luckyc1423
But I am at the average as far as grades for the course so I guess im not the only one struggling in the class.
If it is your first time doing proofs, perhaps that is typical. It is also probably by far the most difficult class(es) out of all courses colleges offer.

Why did you choose a math class instead of something else.

10. I am a math major, I have done extremely well in my other math classes and I have never gotten a C in any. Proofs just seem confusing to me for some reason.

11. Originally Posted by luckyc1423
I am a math major, I have done extremely well in my other math classes and I have never gotten a C in any. Proofs just seem confusing to me for some reason.
i feel your pain Lucky, its the same for me. making the leap from computational math to formal math is a big jump for people like us