I recently attempted to prove the following statement:
"If $x$ and $y$ are real numbers, with $x \ < \ y$ , there is a rational number $r$ such that $x \ < \ r \ < \ y$"
This problem came from a book on proofs that I am currently reading. The author equips the reader with the additional information that...
consider that the following proof is true:
"For all real numbers $\epsilon > 0$ and $a > 0$, there is an integer $n>0$ such that $\frac {a}{n} < \epsilon$"
So, long story short, I came up with the proof and the crucial bit that I developed, which allows me to conclude that the initial statement is true is that:
"$n*\epsilon > 1$"
I will show you the solution manual's proof...I had all of the exact steps besides the aforementioned one:
As you can see, the author decides to state the following:
"$n*\epsilon > 2$"
Now, the purpose of this line (in general) is to establish that you can multiply $x$ and $y$ by some integer that is sufficiently large to ensure that at least one integer will exist between $n*x$ and $n*y$.
However, the I am confused as to why the author decided to use the number $2$ and not $1$. Was the decision arbitrary? Am I missing something?
As far as I can tell, as long as the subtractive difference between two numbers is greater than $1$, you are guaranteed to find an integer that exists between the two numbers.
Would my version of the proof have been equally valid?
