Let $A$ be Lebesgue measurable and $0<\lambda(A)<\infty$. Let $\alpha\in(0,1)$. Prove that there exists an open interval $P$ such that: $$\lambda(A\cap P)\leq\alpha\lambda(P)$$
I found a proof in the internet, but it is for the different inequality ($\geq$). Is this inequality correct, how should I proceed?