Problem #PRU-105063

Problem

The grasshopper jumps on the interval $[0,1]$. On one jump, he can get from the point $x$ either to the point $x/3^{1/2}$, or to the point $x/3^{1/2}+(1-(1/3^{1/2}))$. On the interval $[0,1]$ the point $a$ is chosen.

Prove that starting from any point, the grasshopper can be, after a few jumps, at a distance less than $1/100$ from point $a$.