Agreement of the quantified real-valued definition with the later uniform-metric and uniform-topology formulations

For bounded real-valued functions on a nonempty set, the quantified condition of Pointwise convergence, uniform convergence, and the uniformly Cauchy condition for sequences of real-valued functions agrees with convergence in the supremum metric of The supremum metric $d_\infty(f,g) = \sup_x |f(x) - g(x)|$ is a metric on the bounded real-valued functions on a nonempty set. Indeed, uniform error below $\varepsilon$ gives supremum distance at most $\varepsilon$, while uniform error below $\varepsilon/2$ gives supremum distance strictly below $\varepsilon$; the converse follows because every pointwise error is at most the supremum distance.

The general function-space formulation is Uniform convergence, and the topology of uniform convergence: the metric topology of the uniform metric on $Y^{X}$ and on $C(X,Y)$ ↗, with its convergence dictionary Convergence in the uniform metric is exactly uniform convergence: one $N$ serving every point ↗. The metric-target uniform limit theorem A uniform limit of continuous functions is continuous, so $C(X,Y)$ is closed in $Y^{X}$ under the uniform metric ↗ and the compact-metric version of Dini's theorem Dini's theorem: on a compact metric space a nondecreasing sequence of continuous real functions converging pointwise to a continuous limit converges uniformly ↗ extend the real-valued results proved here. These links are included only for orientation.