MathDB

For each positive real number

r

, define

a_0(r) = 1

and

a_{n+1}(r) = \lfloor ra_n(r) \rfloor

for all integers

n \ge 0

. (a) Prove that for each positive real number

r

, the limit

L(r) = \lim_{n \to \infty} \frac{a_n(r)}{r^n}

exists. (b) Determine all possible values of

L(r)

r

varies over the set of positive real numbers. Here

\lfloor x \rfloor

denotes the greatest integer less than or equal to

x

A4