Getting started on proving this

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Homework Statement



6a. Prove that if 0 is less than or equal to x which is less than y, then x^n < y^n, n = 1, 2, 3...

b. Prove that if x < y and n is odd, then x^n < y ^n.

c. Prove that if x^n = y^n and n is odd, then x = y.

d. Prove that if x^n = y^n and n is even, then x = y or x = -y.

Homework Equations





The Attempt at a Solution



My problem is I'm so confused no how to start this proving something for all n. My attempt at 6a is 6a. I'm not sure how I would do this but my start: x^n < y^n. N = 1, so x < y. If n is two, you can start with x < y. Then you square both sides to get x^n < y^n. Etc. But...how do I prove this for all n?

Logically, I understand why b-d is true, but I have no idea how to prove this. I can probably prove this for specific numbers, but how do I do this for all n? Can someone please provide me with some assistance? I'm supposed to hand in a problem explaining this tomorrow.
 

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LCKurtz
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Homework Statement



6a. Prove that if 0 is less than or equal to x which is less than y, then x^n < y^n, n = 1, 2, 3...

b. Prove that if x < y and n is odd, then x^n < y ^n.

c. Prove that if x^n = y^n and n is odd, then x = y.

d. Prove that if x^n = y^n and n is even, then x = y or x = -y.

Homework Equations





The Attempt at a Solution



My problem is I'm so confused no how to start this proving something for all n. My attempt at 6a is 6a. I'm not sure how I would do this but my start: x^n < y^n. N = 1, so x < y. If n is two, you can start with x < y. Then you square both sides to get x^n < y^n. Etc. But...how do I prove this for all n?

Logically, I understand why b-d is true, but I have no idea how to prove this. I can probably prove this for specific numbers, but how do I do this for all n? Can someone please provide me with some assistance? I'm supposed to hand in a problem explaining this tomorrow.
To prove it by induction the first step is to check whether the statement:

If 0 <= x < y then x^n < y^n

is true for n = 1. That statement is:

If 0 <= x < y then x^1 < y^1, which is obviously true.

I think setting up the induction step is what is bothering you. For the induction step you must show that if the statement is true for n = k then it is true for n = k + 1. So to do the induction step you assume the statement is true for n = k, so you are given:

(*) If 0 <= x < y then x^k < y^k

and what you want to prove is that it is true for n = k + 1:

(**) If 0 <= x < y then x^(k+1) < y^(k+1)

So see if you can use that fact that x < y and that (*) is assumed to be true to get the step that (**) must then be true.
 

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