Negate the following sentence: $\forall x, |x−a| < δ \Rightarrow |f(x)−L| < \varepsilon$

Question

Negate the sentence: $\forall x, |x−a| < δ \Rightarrow |f(x)−L| < \varepsilon$

For my negation I got:

$\exists x, |x-a| < δ \Rightarrow |f(x)-L| \geq \varepsilon$

Would that be correct?

Answer 1

You are correct that $\lnot(\forall x P(x))$ is $\exists x \lnot P(x)$, but your negation of the implication is incorrect. The negation of $P \implies Q$ is $P \land \lnot Q$. See for example this truth table:

 P    Q    P ⇒ ¬Q    P ∧ ¬Q
============================
 T    T      F         F    
 T    F      T         T    
 F    T      T         F     
 F    F      T         F     

In this case, the negation should read in plain English as something like "there is an $x$ such that $x$ is $\delta$-close to $a$ and $f(x)$ is at least $\varepsilon$ away from $L$."