How can we use quantifiers to express something a bit more complex such as - for example - the definition of limit:
$$\forall \epsilon >0\hspace{2mm} \exists \delta >0 \hspace{.5mm} :\hspace{.5mm} 0<|x-a|<\delta \implies |f(x)-L|<\epsilon$$
It's not clear to me how to express double quantifiers and the "such that". My objective is to be able to verify negations of such expressions.
This can be done as follows, adding a quantifier on $x$.
ForAll[\[Epsilon], \[Epsilon] > 0, Exists[\[Delta], \[Delta] > 0,
ForAll[x, Implies[0 < RealAbs[x - a] && RealAbs[x - a] < \[Delta],
RealAbs[f[x] - L] < \[Epsilon]]]]]
For example,
a = 2; f[x_] := x^2; L = 4;
Resolve[ForAll[\[Epsilon], \[Epsilon] > 0,
Exists[\[Delta], \[Delta] > 0, ForAll[x,
Implies[0 < RealAbs[x - a] && RealAbs[x - a] < \[Delta],
RealAbs[f[x] - L] < \[Epsilon]]]]], Reals]
True
If we replace L=4; by L=5;, the above results in False.
