In this document it is shown that the set \(\mathbb{Z}+p\mathbb{Z}\), where \(\mathbb{Z}\) is set of integers and \(p\) is an irrational number, is dense in \( \mathbb{R}\). Not only that but also an equivalent condition that if the set \(\mathbb{Z}+p\mathbb{Z}\) is dense in \( \mathbb{R}\) then \(p\) is irrational. This might be a simple exercise for some but, nonetheless, important as it serves as an example of two closed set whose sum is not closed.

Denseness of m+np in R

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