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The set of natural numbers $\mathbb{N}$ are partitioned into a finite number of subsets.Prove that there exists a subset of $S$ so that for any natural numbers $n$,there are infinitely many multiples of $n$ in $S$.

Hint Short Solution It can also be proved by contradiction as finite×finite not equals to infinite.

May be this answer is not correct as the question asked to prove that there exists a subset $S$ such that in $S$ there are infinitely many multiples of any natural number $n$.

here you have proven for a natural number $n$ there is a subset which have infinite multiple of $n$. But you have to prove in subset $S$ there are infinity many multiples of any natural number $n$