Problem with an alternative (wrong) definition of a limit

Question

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

Answer 1

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.