Two Proofs for Statements a) and b) | Real Numbers, Exponential Inequalities

[TheFerruccio]

I made two attempts at proofs. I feel the second one is ok, but the first one feels lacking. I'm not sure if I could represent it in a better way.

Homework Statement

Prove the following statements

Homework Equations

a) If $$x$$ is real, and $$x > 1$$, then $$x^n > 1$$
b) If $$x$$ is real, and $$x > 1$$, then $$x^m < x^n$$ with $$m < n$$

The Attempt at a Solution

a) Using induction:
1: assume true for n=1
$$x > 1, x^1 > 1$$ since $$x^1 = x$$
2: Assume true for n=k, let $$x^k > 1$$
$$x^{k+1} = x\bullet x^k$$
We know that $$x > 1$$, so $$x^k > x \forall k > 1$$
so $$x x^k$$ is a number greater than 1, multiplied by another number greater than 1, so $$x^{k+1} > 1$$

Therefore, by the principle of mathematical induction... original statement

b) if $$m < n$$ then there exists some integer k such that $$m+k=n$$, $$x^n = x^{m+k}$$, $$\frac{x^n}{x^m} = \frac{x^{m+k}}{x^m} = \frac{x^m x^k}{x^m} = x^k$$, since $$x > 1$$, x^k > 1, so if $$\frac{x^n}{x^m} > 1$$, then $$x^n > x^m$$

 

[eczeno]

the logic behind each proof is sound. so, if that is all you care about then you are good. however, they are worded poorly (and wrongly in at least one place).

the statements themselves are not true if we allow m and n to be a negative integers. it must be stated that n,m are positive integers.

The one thing that makes the first proof actually wrong is that you don't get to assume that the statement is true for k=1 in mathematical induction; it must be shown. but since the k=1 case is stated explicitly in the hypothesis (namely that x>1) there is no work to showing it. (also pick n or k an stick with it during the proof, in the second part you can just say 'assume x^k > 1' and then show that x^(k+1) > 1).

that is the only thing that must be fixed. but the rest of it could use some polishing if you want it to be 'pretty'.

hope this helps.

 

[eczeno]

also, now that i reread your first proof, you use the theorem: if x>1 and y>1 then xy>1. this is of course true, but is a similar statement to the one you are trying to prove and so turning that into a lemma with a quick proof of it on the side would really spiff things up.

again, this is basically aesthetics.