A Partition of $[0, 1)$

Question

Consider a sequence of disjoint nonempty intervals $[\ell_{n}, r_{n}) \subseteq [0, 1)$, $n \in \mathbb{N}$, such that $$\sum_{n = 1}^{\infty} (r_{n} - \ell_{n}) = 1,\qquad 0 < r_{n + 1} - \ell_{n + 1} \leq r_{n} - \ell_{n} < 1. $$ My question is, need the set $$A=[0, 1) \setminus \bigcup_{n \in \mathbb{N}} [\ell_{n}, r_{n})$$ be countable? All constructions I've been able to come up with are such that $A$ is countable. Obviously, $A$ has Lebesgue measure $0$, but that does not imply that $A$ is countable.

Answer 1

HINT: No, because it may be obtained as a small modification of the Cantor set construction.

Answer 2

Consider $l_n=0$ and $r_n=1/(2^n)$ then $\sum_1^\infty r_n=1$ but $[0,1]\setminus \sum_1^ \infty [l_n,r_n)=[1/2,1)$ which is uncountable.Please check it