I found this question in an examination, questions like these are barely covered in my unit, so I'm quite stumped as to how to address them.
This was a two-part question, and the first asked to show $\alpha = \sqrt{4n+2}$ is irrational ($n$ is a fixed positive integer). I understood this and can provide my solution if it helps, but I am unsure to do this second part:
Let ${\{\beta\}}$ denote the 'fractional' part of $\beta$, so $\beta = k + \{\beta\}$, where $k$ is an integer and $\le \{\beta\} < 1$.
Let $N$ be a positive integer. Consider the numbers: $0, \{\alpha\}, \{\alpha\}, .... \{(N-1)\alpha\}, \{N\alpha\}$.
By dividing the interval $[0,1]$ into sections, or otherwise, prove that at least two of these numbers differ by less than $1/N$.
Any explanation/assistance is greatly appreciated.