Proving Exponential Inequalities for Positive Real Numbers

[julypraise]

(Sorry for the wrong title!)

Let a>1 (a in R). Then how do you prove for all x in R, a^x > 0 ?

And also that a^y>1 for all y>0, how do you prove?

And also how do you prove that 0^b = 0 when b is a real number?

(For me this is so difficult. So please just help me. Enough hints will suffice, no full solution.)

 

[micromass]

This depends on what you can use. I would prove it by the following scheme:

1) First prove it for all natural numbers x (by induction).

2) Use a^{-x}=(a^x)^{-1} to prove it for all integers.

3) Use a^{m/n}=\sqrt[n]{a^m} to prove it for all rational numbers.

4) Use continuity of a^x to prove it for all real numbers.

 

[julypraise]

Could you help me bit more on proving the part 3)?

So by the theorem in Rudin's Mathematical Analysis in page 10 thm 1.21, I can prove this right?

 

[checkitagain]

Third problem:

And also how do you prove that 0^b = 0 when b is a real number?

b must be restricted, as in b > 0, so that 0^b is not

undefined and/or is not indeterminate.