Real Analysis like convergence of loss functions

Here's one thing i noticed. In Elementary Real Analysis, when we say that a sequence $$s_n$$ converges to a point $$s$$, we first set an $$\epsilon > 0$$ such that for large $$N \in \mathbb{N}$$ $$| s_n - s | < \epsilon.$$

Why don't we set up the convergence of loss functions that way? i. e. We don't have a hyperparameterhyperparameter $$\epsilon$$ to ensure our convergence.