For an introduction to young audiences, I think it’s better to get the point across in less technical terms before trying to formalize it. The OP jumps to epsilon pretty quickly. I would try to get to a description like “A sequence converges to a limit L if its terms are ‘eventually’ arbitrarily close to L. That is, no matter how small a (nonzero) tolerance you pick, there is a point in the sequence where all of the remaining terms are within that tolerance.” Then you can formalize the tolerance, epsilon, and the point in the sequence, k, that depends on epsilon.
Note that this doesn’t depend on the sequence being indexed by integers or the limit being a real number. More generally, given a directed set (S, ≤), a topological space X, and a function f: S → X, a point x in X is the limit of f if for any neighborhood U of x, there exists t in S where s ≥ t implies f(s) in U. That is, for every neighborhood U of x, f is “eventually” in U.
For an introduction to young audiences, I think it’s better to get the point across in less technical terms before trying to formalize it. The OP jumps to epsilon pretty quickly. I would try to get to a description like “A sequence converges to a limit L if its terms are ‘eventually’ arbitrarily close to L. That is, no matter how small a (nonzero) tolerance you pick, there is a point in the sequence where all of the remaining terms are within that tolerance.” Then you can formalize the tolerance, epsilon, and the point in the sequence, k, that depends on epsilon.
Note that this doesn’t depend on the sequence being indexed by integers or the limit being a real number. More generally, given a directed set (S, ≤), a topological space X, and a function f: S → X, a point x in X is the limit of f if for any neighborhood U of x, there exists t in S where s ≥ t implies f(s) in U. That is, for every neighborhood U of x, f is “eventually” in U.