Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities

Setting. Let $\mathbb{R}$ be the real numbers, an ordered field with order $\le$; for $s,t\in\mathbb{R}$ write $s<t$ for the associated strict order ($s\le t$ and $s\ne t$), $|x|$ for the absolute value, and $x^{n}$ ($n$ a natural number) for the natural number power. Let $\mathbb{N}$ be the set of natural numbers with the order $\le$. Natural numbers are regarded as real numbers through the canonical map $\iota:\mathbb{N}\to\mathbb{R}$, which is suppressed from the notation (the convention of clause 3 of The Real Numbers and Standard Notation); by claims 2 and 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field this map is strictly increasing with $\iota(n)\ge1$ for every $n\in\mathbb{N}$, so that $n\ge1$ for every $n\in\mathbb{N}$ and inequalities between natural numbers may be read in $\mathbb{N}$ or in $\mathbb{R}$ indifferently (the passage from $\mathbb{R}$ back to $\mathbb{N}$ using that the order on $\mathbb{N}$ is total); "$n\in\mathbb{N}$" for a real number $n$ means that $n$ is the image of a natural number. The integers enter only through the integer part. Let $\exp$ be the exponential function, $\log:(0,\infty)\to\mathbb{R}$ the natural logarithm, $\sqrt{a}$ the nonnegative square root of a real $a\ge0$, and $\lfloor x\rfloor$ the integer part of a real $x$. Sequences of real numbers are indexed by $\mathbb{N}$, and their limits are as in that definition; $a_{N}\to L$ means that $(a_{N})_{N\in\mathbb{N}}$ converges to $L$.

Real powers. For a real number $t>0$ and a real number $a$ let $t^{a}=\exp(a\log t)$ be the real power of $t$ with exponent $a$. For a natural number $N$ the real power $N^{a}$ is formed with $N$ regarded as a positive real number. Where an exponent is a natural number, or is $\tfrac12$, the symbol $t^{a}$ could also be read as the natural number power or as $\sqrt{t}$; claim 1 shows that all readings agree.

Then the following hold.

1. (Algebra of powers.) Let $s,t>0$ and $a,b\in\mathbb{R}$. Then $t^{a}>0$; $t^{0}=1$ and $t^{1}=t$; $t^{a+b}=t^{a}t^{b}$ and $t^{-a}=1/t^{a}$; $(st)^{a}=s^{a}t^{a}$; $\log(t^{a})=a\log t$ and $(t^{a})^{b}=t^{ab}$; for every natural number $n$ the real power $t^{n}$ equals the natural number power $t^{n}$, and $(t^{a})^{n}=t^{na}$ in either sense; and $t^{a/2}=\sqrt{t^{a}}$, in particular $t^{1/2}=\sqrt{t}$, $t^{1/4}=\sqrt{\sqrt{t}}$, $t^{1/8}=\sqrt{t^{1/4}}$ and so on, and $t^{-a}=1/t^{a}$ shows that $t^{-1/2}=1/\sqrt{t}$ and $t^{-1/4}=1/\sqrt{\sqrt{t}}$.

2. (Monotonicity.) $\log$ is strictly increasing on $(0,\infty)$, $\log1=0$, and for $t>0$ one has $\log t\ge0$ if and only if $t\ge1$. Let $a>0$. If $0<s\le t$ then $s^{a}\le t^{a}$, and if $0<s<t$ then $s^{a}<t^{a}$; if $t\ge1$ and $b\le c$ are real then $t^{b}\le t^{c}$, in particular $t^{b}\ge1$ for $b\ge0$; if $0<t\le1$ and $b\le c$ then $t^{b}\ge t^{c}$. Moreover, for $t>0$ and $x>0$: $t^{a}\ge x$ if and only if $t\ge x^{1/a}$; $t^{a}>x$ if and only if $t>x^{1/a}$; and $t^{-a}<x$ if and only if $t>x^{-1/a}$.

3. (Null sequences and eventual bounds.) (a) For every real $a>0$ the sequence $(N^{-a})_{N\in\mathbb{N}}$ converges to $0$. (b) For all real $a>0$, $c>0$ and $k\ge0$ the sequence $\bigl(N^{k}\exp(-cN^{a})\bigr)_{N\in\mathbb{N}}$ converges to $0$; more precisely, if $m\in\mathbb{N}$ satisfies $am\ge k$, then $0\le N^{k}\exp(-cN^{a})\le(m+1)^{m+1}c^{-(m+1)}N^{-a}$ for every $N\in\mathbb{N}$. (c) If $(c_{N})$ and $(b_{N})$ are real sequences, $L$ is real, $N_{0}\in\mathbb{N}$, $|c_{N}-L|\le b_{N}$ for every $N\ge N_{0}$, and $b_{N}\to0$, then $c_{N}\to L$. (d) If $c_{N}\to L$ and $x$ is real with $L<x$, then there is $N_{0}\in\mathbb{N}$ with $c_{N}<x$ for every $N\ge N_{0}$; if $L>x$, then there is $N_{0}$ with $c_{N}>x$ for every $N\ge N_{0}$. (e) If $a_{N}\ge0$ for every $N$ and $a_{N}\to A$, then $A\ge0$ and $\sqrt{a_{N}}\to\sqrt{A}$; consequently also $\sqrt{\sqrt{a_{N}}}\to\sqrt{\sqrt{A}}$. (f) If $a_{N}\to A$ then $\exp(a_{N})\to\exp(A)$. (g) If $a_{N}\to A$ with $A>0$ and $a_{N}>0$ for every $N$, and $b$ is real, then $a_{N}^{\,b}\to A^{b}$.

4. (Integer rounding.) For every real $x$ the real number $n=\lfloor x\rfloor+1$ satisfies $x<n\le x+1$; if $x\ge0$ then $n\in\mathbb{N}$, that is, $n$ is the image of a natural number. Consequently, for every real $x$ there is $N_{0}\in\mathbb{N}$ with $N>x$ for every natural number $N\ge N_{0}$.

5. (Square roots.) Let $a,b\ge0$ be real. Then $\sqrt{ab}=\sqrt{a}\sqrt{b}$ and $\sqrt{a^{2}}=a$; if $a\le b$ then $\sqrt{a}\le\sqrt{b}$, and if $a<b$ then $\sqrt{a}<\sqrt{b}$; $\sqrt{a+b}\le\sqrt{a}+\sqrt{b}$; $|\sqrt{a}-\sqrt{b}|\le\sqrt{|a-b|}$; $\sqrt{a}\le1+a$; and $(a+b)^{2}\le2a^{2}+2b^{2}$.

6. (Exponential inequalities.) For every real $x\ge0$,

and for every real $h$, $|\exp(h)-1|\le|h|\exp(|h|)$; moreover $\exp(-x)\le1/(1+x)\le1$ for $x\ge0$, and $\exp$ is nondecreasing, so that $\exp(-u)\le\exp(-v)$ whenever $u\ge v$.