convergence in a metric space
A sequence converges to a point when its terms eventually huddle as close to that point as you like and stay there. Convergence in a metric space is the same idea as on the real line, with the absolute value replaced by the distance d, so the notion travels intact to spaces of functions, vectors, or anything else with a metric.
Precisely, a sequence (x_n) in a metric space (X, d) converges to a limit L if for every epsilon > 0 there is an N such that d(x_n, L) < epsilon for all n >= N. Equivalently, the real sequence d(x_n, L) tends to 0. The condition is exactly the epsilon-N definition of convergence with |x_n - L| swapped for d(x_n, L).
Limits are unique whenever the space is a genuine metric space, because if a sequence approached two limits L and L-prime, the triangle inequality would force d(L, L-prime) <= d(L, x_n) + d(x_n, L-prime) -> 0, so L = L-prime. Crucially, whether a sequence converges can depend on the metric: the same sequence of functions may converge under one notion of distance and diverge under another.
In R^2 with Euclidean distance, the sequence x_n = (1/n, 1 - 1/n) converges to (0, 1): the distance d(x_n, (0,1)) = sqrt(1/n^2 + 1/n^2) = sqrt(2)/n -> 0.
Convergence reduces to the real sequence of distances going to 0.