Question

Prove that a subset of a countably infinite set is finite or countably infinite.

Answer 1

Lok s loe a Coun-tabla in-finire or, enumerable Sat in fi mi te set, is Coun ta Since Com ba de sori bed as a, a con lains infinite number f a's owd the am infinte Suv setP o IN Then the orn PoP 0 lem on Sa 1 Con tains a lea infit Cula set oS aut the Su fixes of IN Con toi ns a Or is asain an infinite Subset of s. Proceeding with si mlar aィ0umenks-, ube with Sim la r H344

Let リsda: fine- mapping a 0 1c (n)a prove that f is injec tive, let- C- a. (a) ap n-1 (Cp)メーf (1) and -this proves that s mictive natural number γ, -There aYe at mest (3-1) elem en Hsin ㅜ who se- "suffixes are less than γ. soar is one -((2), .((T) リク , . there tare is Su ec thive thus İs -T oun tobaly enumera ble ,or coun +olong nino s in Hen ce, ubset. of a can con clude that, A s wa- eithrífiniteov co u n tably Coun tobn n 'n finite com to big ns Set- ininiFe.