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 $b\geq1$.

1. If $m,n,p,q$ are integers, $n\geq0,q\geq0$, 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 $y$.

It is a litter hard for beginners to solve this problem, so I will try to post my solution here. For a. suppose $\alpha=(b^m)^{1/n}$ and $\beta=(b^p)^{1/q}$, then by the definition of $n$th roots, we know that \[ \alpha=(b^m)^{1/n}\Leftrightarrow \alpha^n=b^m, \quad \text{and}\quad \beta=(b^p)^{1/q}\Leftrightarrow \beta^q=b^p. \] Thus, by the meaning of notation $b^m$ (which says $b$ multiples itself by $m$ times), we know that \[ (b^m)^p=b^{mp}=(b^p)^m, \] thus \[ \alpha^{np}=(\alpha^n)^p=(b^m)^p=(b^p)^m=(\beta^q)^m=\beta^{qm}, \] since $r=m/n=p/q$, we have $pn=mq$, and \[ \alpha=\beta, \] by the uniqueness of $n$th roots of a real number. That is $(b^m)^{1/n}=(b^p){1/q}$. For b. we can assume $r=m/n$ and $s=p/q$ for $n,q>0$ and $m,n,p,q$ are integers. Then by a., \[ b^{r+s}=b^{\frac{m}{n}+\frac{p}{q}}=b^{\frac{mq+np}{nq}}=(b^{mq+np})^{1/nq}=(b^{mq}b^{np})^{1/nq} =(b^{mq})^{1/nq}(b^{np})^{1/nq}=b^{m/n}b^{p/q}=b^rb^s, \] we employ the result (prove in the book) that: \[ (\alpha\beta)^{1/n}=\alpha^{1/n}\beta^{1/n}, \] for any $\alpha,\beta\in\R$ and $n$　be a positive integer. Now, for c., by the definition, we can write $B(x)$ as \[ B(x)=\set{b^t|t\in Q, t\leq x}, \] and note that $b>1$ thus \[ b^t\leq b^r, \] for all $t,r\in Q$ with $t\leq r$. In fact, if $b^t>b^r$. Set $t=m/n$, $r=p/q$, and $t\leq r$ then by the definition of ordered field, \[ mq\leq np\Rightarrow b^{mq}\leq b^{np}, \] thus by a. \[ (b^t)^{nq}=b^{mq}\leq b^{np}=(b^{r})^{nq}, \] a contradiction to the assumption $b^t>b^r$. In conclusion, we know that \[ \sup \set{b^t|t\in Q, t\leq r}\leq b^r, \] and apparently, \[ b^r\leq\sup B(r)=\sup \set{b^t|t\in Q, t\leq r}, \] thus $b^r=\sup B(r)$. For d., firstly, for any $r,s\in Q$, if $r\leq x$, $s\leq y$, then by the definition of $b^x, b^y$, we have \[ b^r\leq b^x,\quad b^s\leq b^y, \] thus \[ b^{r+s}=b^rb^s\leq b^xb^y, \] since $\R$ is a ordered field. Now, for any $t\in Q$, if $t\leq x+y$, we can always find $r,s\in Q$ such that $r\leq x, s\leq y$ and $r+s=t$. In fact, by the definition of $x+y$, we know that $x+y=\set{r+s|r\in x, s\in y}$, thus the assertion is certainly satisfied. Consequently, \[ b^{x+y}=\sup\set{b^t|t\in Q, t\leq x+y}\leq b^xb^y. \] On the other hand, if $b^{x+y}< b^xb^y$, then set $\varepsilon=b^xb^y-b^{x+y} >0$. For any small enough $\delta>0$, by the definition of $b^x, b^y$, there exist $r_0,s_0\in Q$, with $r_0\leq x$, $s_0\leq y$, such that \[ b^{r_0}>b^x-\delta>0,\quad b^{s_0}>b^y-\delta>0. \] Since $\R$ is an ordered field, then \[ b^{r_0}b^{s_0}>b^xb^y-\delta\left(b^x+b^y-\delta\right), \] particularly, we can take $\delta$ be the positive roots of \[ \delta\left(b^x+b^y-\delta\right)=\eps/2, \] then \[ 2b^{x+y}\geq 2b^{r_0}b^{s_0} >2b^xb^y-\eps=b^xb^y+b^{x+y}, \] that is \[ b^{x+y}>b^xb^y, \] contradict to the assumption.