How tounderstand the proof to show if a real number $w>0$ and a real number $b>1$, $b^w>1$? Claim: If there are two real numbers $w>0$ and $b>1$, then $b^w>1$. 
One proof is that for any rational number $0<r<w$ and $r=m/n$ with $m,n\in\mathbb Z$ and $n\ne0$, $(b^m)^{1/n}>1$. To show $b^m>1$, you can use induction, and then use $1^n=1<b^m$ to finally show that $b^r>1$. So the proof is done. 
I do not think this proof is complete. If $w$ is also a rational number, everything is fine. But if $w$ is an irrational number, I think the proof is incomplete, but I am not sure how to proceed. 
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Please do not use the continuity or more advanced techniques. Assume you just finished the first chapter of Rudin's Principles of Mathematical Analysis. 

If
$b > 1$,
then
$b = 1+c$
where
$c > 0$,
so
$b^n
=(1+c)^n
\ge 1+nc
\gt 1
$
by
Bernoulli's inequality.
If
$b^{n/m} = a$,
then
$b^n = a^m$.
Since
$b^n > 1$,
$a^m > 1$
so that
$a > 1$.
Then apply the definition
 $b^x
=\sup_{r \le x} b^r
$.
