for all and for every , what's wrong with this simple contradiction? For these two statements,
(1). There exists a positive real number $\dfrac {\varepsilon } {2}$
    smaller than every  positive real number $ {\varepsilon } $ .
(2). There is no positive real number smaller than every  positive real
        number $ {\varepsilon } $ .
Questions :


*

*(1) contradicts with (2),  which is wrong and which is right ?

*What I got confused to get this contradiction?

 A: The order of your logical symbols is important.


*

*For all $\epsilon > 0$ there exists a smaller real number. 
$$
\forall\epsilon>0\exists\delta>0 \text{ such that } \delta<\epsilon
$$

*There does not exists a number that is smaller for all $\epsilon>0$. 
$$
\nexists \delta > 0 \forall \epsilon>0 \text{ such that } \delta<\epsilon
$$


Note that these statements don't contradict eachother. Actually, they are the same statement. Call the second statement $A$. Then $A = (A^c)^c$. 
\begin{align}
A^c &= \exists \delta > 0 \forall \epsilon>0 \text{ such that } \delta<\epsilon. \\
(A^c)^c &= \forall\delta > 0 \exists\epsilon>0 \text{ such that } \delta>\epsilon
\end{align}
This is the same as statement 1 with $\epsilon$ and $\delta$ reversed. 
A: The subtlety here fooled me once to.
The key to understanding the contradiction is noticing what is fixed first. 
For "There exists a positive real number $\dfrac {\varepsilon } {2}$
    smaller than every  positive real number $ {\varepsilon } $ " we first set $\epsilon $, any epsilon, and then say $\dfrac {\varepsilon } {2}$ is smaller, which is true.
For "There is no positive real number smaller than every  positive real
        number $ {\varepsilon } $" we are saying that, there is no number that we can fix that will be smaller than any other. They are saying that whatever, for how many small, $x $ is, there will be a smaller $\epsilon $.
