proof of growth of exponential function

In this proof, we first restrict to when $x$ and $a$ are integers and only later lift this restricton.

Let $a>0$ be an integer, let $b>1$ be real, and let $x$ be an integer.

Consider the following inequality

 $\left(1+{1\over x}\right)^{a}\leq 1+{a\over x}\left(1+{1\over x}\right)^{a-1}$

If $x\geq 2$, then we have

 $\left(1+{1\over x}\right)^{a}\leq 1+{a\over x}\left({3\over 2}\right)^{a-1}.$

Define $X$ to be the greater of $2$ and $\lceil a(3/2)^{a-1}/(1-\sqrt{b})\rceil$; when $x>X$, we have

 $\left(1+{1\over x}\right)^{a}\leq\sqrt{b}.$

Rewrite $x^{a}/b^{x}$ as follows when $x>X$:

 ${x^{a}\over b^{x}}={X^{a}\over b^{X}}\prod_{n=X}^{x}\left(1+{1\over n}\right)^% {a}{1\over b}$

By the inequality established above, each term in the product   will be bounded by $1/\sqrt{b}$, hence

 ${x^{a}\over b^{x}}\leq{X^{a}\over b^{X}}{1\over(\sqrt{b})^{x-X}}$

Since $b>1$, it is also the case that $\sqrt{b}>1$, hence we have the inequality

 $(\sqrt{b})^{n}\geq 1+n(\sqrt{b}-1)$

Combining the last two inequalities yields the following:

 ${x^{a}\over b^{x}}\leq{X^{a}\over b^{X}}\leq{1\over 1+(x-X)(\sqrt{b}-1)}$

From this, it follows that $\lim_{x\to\infty}x^{a}/b^{x}=0$ when $a$ and $x$ are integers.

Now we lift the restriction   that $a$ be an integer. Since the power function is increasing, $x^{a}/b^{x}\leq x^{\lceil a\rceil}/b^{x}$, so we have $\lim_{x\to\infty}x^{a}/b^{x}=0$ for real values of $a$ as well.

To lift the restriction on $x$, let us write $x=x_{1}+x_{2}$ where $x_{1}$ is an integer and $0\leq x_{2}<1$. Then we have

 ${x^{a}\over b^{x}}={x_{1}^{a}\over b^{x_{1}}}\left({x_{1}+x_{2}\over x_{1}}% \right)^{a}b^{-x_{2}}$

If $x>2$, then $(x_{1}+x_{2})/x_{2}<1.5$. Since $x_{2}\geq 0,b^{-x_{2}}\leq 1$. Hence, for all real $x>2$, we have

 ${x^{a}\over b^{x}}\leq 1.5^{a}{x_{1}^{a}\over b^{x_{1}}}$

From this inequality, it follows that $\lim_{x\to\infty}x^{a}/b^{x}=0$ for real values of $x$ as well.

Title proof of growth of exponential function ProofOfGrowthOfExponentialFunction 2013-03-22 15:48:36 2013-03-22 15:48:36 rspuzio (6075) rspuzio (6075) 11 rspuzio (6075) Proof msc 32A05