Prove that if B A and A is countable then B is countableSolu

Prove that if B A and A is countable, then B is countable.

Solution

Let A be a countable set and B A.

Suppose a₁, a₂ ,a₃, ... is an enumeration of the countable set A and B is any nonempty subset of A.

If, for some n N the element a_n B then we assign the natural number n to it. For each n N , let k(n) denote the number of elements among a₁, a₂ ,a₃, ... a_n which belong to the subset B .

Then 0 k(n) n Therefore, B is countable by the Countability Lemma

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