BALTIC-WAY 2020 Problem 10

Alice and Bob are playing hide and seek. Initially, Bob chooses a secret fixed point $B$ in the unit square. Then Alice chooses a sequence of points $P_{0}, P_{1}, \ldots, P_{N}$ in the plane. After choosing $P_{k}$ (but before choosing $P_{k+1}$ ) for $k \geqslant 1$, Bob tells "warmer" if $P_{k}$ is closer to $B$ than $P_{k-1}$, otherwise he says "colder". After Alice has chosen $P_{N}$ and heard Bob's answer, Alice chooses a final point $A$. Alice wins if the distance $A B$ is at most $\frac{1}{2020}$, otherwise Bob wins. Show that if $N=18$, Alice cannot guarantee a win.