MathDB

We can conduct the following moves to a real number

x

: choose a positive integer

n

, and positives reals

a_1,a_2,\cdots, a_n

whose reciprocals sum up to

1

. Let

x_0=x

, and

x_k=\sqrt{x_{k-1}a_k}

for all

1\leq k\leq n

. Finally, let

y=x_n

. We said

M>0

is "tremendous" if for any

x\in \mathbb{R}^+

, we can always choose

n,a_1,a_2,\cdots, a_n

to make the resulting

y

smaller than

M

. Find all tremendous numbers.
Proposed by ckliao914.

Problem Statement