$C \subset \mathbb{R}$ finite, $\lambda(A)>0$. Prove the existence of $a$ and $b \ne 0$ such that $\{a+bx : x \in C\} \subset A$

Question

Let $C = \{c_1, c_2, ... , c_n\} \subset \mathbb{R} $ be a finite subset of $\mathbb{R}$, and let $A \subset \mathbb{R}$ be a set with positive measure $\lambda(A)>0$

Prove that there are $a,b \in \mathbb{R}$, $b \ne 0$, such that

$$a + bC := \{a+bx:x \in C\} \subset A$$

A hint was given:

Find an interval $I$ such that $\lambda(A\cap I) > (1-\varepsilon)\lambda(I)$

It's pretty easy to find such an interval using Lebesgue's density theorem, but I don't know how to continue.