Prove that the cardinality of [0,1] is the same as the cardinality of (0,1) by showing that ∣[0,1]∣≤∣(0,1)∣ and ∣(0,1)∣≤∣[0,1]∣. See Definition 0.3.28. This proof requires the Cantor--Bernstein--Schröder theorem, which we stated without proof. Note that this proof does not give you an explicit bijection.