During my math studies , I encountered variable substitutions as shown in the following examples:
$ \forall \epsilon>0 \, \exists N\in \mathbb{N} \, \forall n \in N . | a_n - L | < \epsilon $
$ \forall \epsilon>0 \, \exists N\in \mathbb{N} \, \forall n \in N . | a_n - L | < \frac{\epsilon}{4} $
Another example:
$ \forall n\in \mathbb{N} \, \exists s \in \mathbb{R} .n \leq s $
$ \forall n\in \mathbb{N} \, \exists s \in \mathbb{R} .n+2 \leq s $
$ \forall n\in \mathbb{N} \, \exists s \in \mathbb{R} .n+4 \leq s $
- How do we go from $ \forall \epsilon>0 \, \exists N\in \mathbb{N} \, \forall n \in N . | a_n - L | < \epsilon $
to: $ \forall \epsilon>0 \, \exists N\in \mathbb{N} \, \forall n \in N . | a_n - L | < \frac{\epsilon}{4} $ ?
My naive thought was just to change the bound variable $ \epsilon $ to $ \frac{\epsilon}{4} $ so as to have:
$ \forall \frac{\epsilon}{4}>0 \, \exists N\in \mathbb{N} \, \forall n \in N . | a_n - L | < \frac{\epsilon}{4} $
But clearly this is incorrect since $ \frac{\epsilon}{4} $ is just a symbol and the 4 in the denominator is just part of the symbol and not really a number.
- How do we go from $ \forall n\in \mathbb{N} \, \exists s \in \mathbb{R} .n \leq s $
to: $ \forall n\in \mathbb{N} \, \exists s \in \mathbb{R} .n+2 \leq s $?
Can you please give a rigorous explanation? ( I understand the intuition behind these representations, but I can't really concisely explain and/or write on a paper how do we go from one representation to another of these logical statements. )
Think of how you would prove that universal statement $ \forall \epsilon>0 \, \exists N\in \mathbb{N} \, \forall n \in N . | a_n - L | < \frac{\epsilon}{4} $. You would let $\epsilon$ be any real number greater than $0$, and you would have to show that for such an $\epsilon$, we have that $\exists N\in \mathbb{N} \, \forall n \in N . | a_n - L | < \frac{\epsilon}{4} $
OK, but why would the latter be true? It is because if $\epsilon > 0$, then we also have that $\frac{\epsilon}{4} > 0$, and we already know that for any real number (let's call it $\gamma$) it is true that $ \forall \gamma>0 \, \exists N\in \mathbb{N} \, \forall n \in N . | a_n - L | < \gamma $. Why do we know that? Because we were given that $ \forall \epsilon>0 \, \exists N\in \mathbb{N} \, \forall n \in N . | a_n - L | < \epsilon $, but the $\epsilon$ is just a 'dummy' variable, i.e. we can use any other variable ... in this case $\gamma$.
So, since that is true for any $\gamma > 0$ that $ \exists N\in \mathbb{N} \, \forall n \in N . | a_n - L | < \gamma $, the latter is specifically true for $\gamma = \frac{\epsilon}{4}$, and thus we know that for any $\epsilon > 0$ it is true that $\exists N\in \mathbb{N} \, \forall n \in N . | a_n - L | < \frac{\epsilon}{4} $.
In other words: $ \forall \epsilon>0 \, \exists N\in \mathbb{N} \, \forall n \in N . | a_n - L | < \frac{\epsilon}{4} $.