Negating nested quantifiers

Negation.
$$\exists\varepsilon>0,\ \forall N\in\mathbb{N},\ \exists n\ge N,\ |u_n-\ell|\ge\varepsilon.$$

Interpretation. There is some $\varepsilon>0$ such that, no matter how far we go, some term of the sequence is at distance $\ge\varepsilon$ from $\ell$. This is exactly the definition of "$(u_n)$ does not converge to $\ell$".

Conclusion. The negation captures non-convergence to $\ell$.