MathDB

Let

A

be a real number and

(a_{n})

be a sequence of real numbers such that

a_{1}=1

and 1<\frac{a_{n+1}}{a_{n}}\leq A \mbox{ for all }n\in\mathbb{N}.

(a)

Show that there is a unique non-decreasing surjective function

f: \mathbb{N}\rightarrow \mathbb{N}

such that

1<A^{k(n)}/a_{n}\leq A

for all

n\in \mathbb{N}

(b)

k

takes every value at most

m

times, show that there is a real number

C>1

such that

Aa_{n}\geq C^{n}

for all

n\in \mathbb{N}

3