- #1
Bashyboy
- 1,421
- 5
Homework Statement
Let ##S = \{\frac{1}{n} + \mathbb{Z} ~|~ n \in \mathbb{N} \}##. I am trying to show that ##f : \mathbb{N} \rightarrow S## defined by ##f(n) = \frac{1}{n} + \mathbb{Z}## is a bijection. Surjectivity is trivial, but injectivity is a little more involved.
Homework Equations
The Attempt at a Solution
Suppose that ##f(n) = f(m)##. Then ##\frac{1}{n} + \mathbb{Z} = \frac{1}{m} + \mathbb{Z}## or ##\frac{1}{n} - \frac{1}{m} \in \mathbb{Z}##. This implies ##\frac{m-n}{nm}## is an integer. If ##m \neq n##, then the fraction is not zero but a ratio of two nonzero integers. However, if ##\frac{p}{q}## is ##\frac{m-n}{nm}## in reduced form, how do I know that ##q = 1## can't be true; how do I know that ##m \neq n## and ##p=1## can't both be true?