A theorem on unit segments on the real line

Let n ≥ 1 be an odd integer. For every 1 ≤ i ≤ n let s_i = (a_i, b_i) be an open unit segment on the real line. Let (Equation presented) be fixed. Color by green all the points (numbers) on the real line of the form a_i + ϵ and b_i − ϵ. Then there exists at least one green point that belongs to an odd number of the segments s₁, …, s_n.