Let $x_1, \ldots, x_n$ be random variables that are uniformly distributed on [0,2]. If $x_i \leq 1$, we'll classify it as green ($y_i=0$), and if $x_i > 1$, we'll classify it as red ($y_i=1$). We want to classify the point $x=1.1$ with the $k$-NN algorithm where $k=1$ and data points $(x_1,y_1),\ldots,(x_n,y_n)$.
Prove that the probability of classifying $x=1.1$ correctly as red converges to 1 as n goes to infinity.
Hint: Do not try to calculate the probabilities.
I haven't been able to solve the above problem. Can you solve it? I've tried to calculate the probabilities even though the hint suggests not to do this. I've also tried using inequalities such as Markov's.