Pointwise vs Uniform Convergence — 2D Trace View

Real analysis draws a sharp line between two ways a sequence of functions can "converge": pointwise, where each individual point settles down at its own pace, and uniform, where a single tolerance works everywhere on the domain at once. This 2D companion view renders the same three textbook function sequences as two linked Cartesian plots instead of a 3D surface: the current curve f_n(x) against its limit with a shaded ε-tolerance band, and — directly below it — a running trace of the sup-norm S(n) = sup_x|f_n(x) − f(x)| against the index n. Watching that second trace either decay to zero and stay there (uniform convergence) or flatten out at a stubborn positive floor (not uniform) makes the abstract ε–N definition of uniform convergence directly visible as a single curve, including the classic case — the travelling bump n·x·(1−x)ⁿ — where the limit is continuous yet convergence still fails to be uniform.