null sequences

...

I don't quite understand the mathematical statement with the three quantifiers is the issue I think

Yes, this is the problem.

The definition says:

$... \forall n \in \mathbb{N}, (n \ge N \implies |f(n)| < \epsilon )$.

Rewriting this to be a bit wordier, we get: "for all natural numbers n, the statement "$n \ge N$" implies |f(n)| < epsilon.

Your interpretation is more like: "the statement "for all natural numbers n, $n \ge N$" implies |f(n)| < epsilon". (If you wrote this with quantifies it wouldn't look like the definition, it would be more like: $... (\forall n \in \mathbb{N}, \,n \ge N) \implies |f(n)| < \epsilon )$).

As far as understanding the definitions goes, it's probably best not to think too much in terms of the quantifier symbols - they save writing, but they don't help clarity.

Instead:

f(n) is null if given $\epsilon > 0$ we can find N such that whenever n is an integer > N, we have $|f(n)| < \epsilon$.

A null sequence is one that converges to zero.

In terms of the definition. Choose any distance from zero (that's our epsilon), then there is a point in the sequence, N, beyond which (n>N) all values, f(n), in the sequence lie within epsilon of zero. |f(n)| < epsilon.