MathDB

For the positive integer

n

, define

f(n)=\min\limits_{m\in\Bbb Z}\left|\sqrt2-\frac mn\right|

. Let

\{n_i\}

be a strictly increasing sequence of positive integers.

C

is a constant such that

f(n_i)<\dfrac C{n_i^2}

for all

i\in\{1,2,\ldots\}

. Show that there exists a real number

q>1

such that

n_i\geqslant q^{i-1}

for all

i\in\{1,2,\ldots \}

Problem Statement