The classic definition for limit L of a sequence $(a_n)_{n=1}^{\infty}$ is to say that it satisfies the following: $(\forall \epsilon >0)(\exists N \in N)(\forall n \in N)(n \geq N \implies |x_n-L|<\epsilon$
My professor just pointed out the other day that the order of the quantifiers is important, and for instance defining the limit L of a sequence in the following way would not be equivalent (and wrong based on the general notion of a limit):
$(\forall n \in N)(\forall \epsilon >0)(\exists N \in N)(n \geq N \implies |x_n-L|<\epsilon$
However, I'm having trouble visualising the difference. What would limit mean according to the second definition? Could anyone come up with a counterexample of a sequence whose limit would change (or be undefined) based on this definition
In the sense of the second definition, every sequence converges to every value. Note that it suffices to pick (given $n$ and $\epsilon$) $N$ as $n+1$, for example. Then $n\ge N$ is false and so the implication vacuously true.