Solution of Rudin’s Principles of Mathematical Analysis:Chap1. Ex.6

The definition of (real) exponent of $b\in\R$, $b>0$ is give in Rudin’s Principles of Mathematical Analysis (3.ed), Chapter 1 Exercise 6 , which says that:

6. Fix $b>1$.

  1. If $m,n,p,q$ are integers, $n>0,q>0$, and $r=m/n=p/q$, prove that
    \[
    (b^m)^{1/n}=(b^p)^{1/q}.
    \]
    Hence it makes sense to define $b^r=(b^m)^{1/n}$.
  2. Prove that $b^{r+s}=b^rb^s$ if $r$ and $s$ are rational.
  3. If $x$ is real, define $B(x)$ to be the set of all numbers $b^t$, where $t$ is rational and $t\leq x$. Prove that
    \[
    b^r=\sup B(r)
    \]
    when $r$ is rational. Hence it makes sense to define
    \[
    b^x=\sup B(x)
    \]
    for every real $x$.
  4. Prove that $b^{x+y}=b^xb^y$ for all real $x$ and $y$.

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