Pointwise Convergence

Real Analysis · Axiom Academy

A sequence of functions converges pointwise when convergence happens independently at each fixed point — possibly at very different rates. 1. The Definition: One Point at a Time Fix a single point x and forget the rest of the domain. All that is left is a plain sequence of numbers f_n(x) . If that sequence converges to f(x) , we have pointwise convergence at x . Convergence of the functions just means: this happens at every x . The value at a fixed point is an ordinary number sequence It must land on f(x) — the height of the limit function there 2. Different Rates at Different Points Pointwise convergence promises that each point eventually settles — but says nothing about doing so at the same speed. Watch the very same sequence f_n(x) = x^n at three fixed inputs: the value reaches the target 0 almost at once near x = 0 , yet crawls there near x = 1 . f_n(0.3) = 0.3^n is already by n = 5 . Converges fast. 0.6^n needs roughly n = 14 to drop below 0.001 . Moderate. 0.9^n takes about n = 66 to reach 0.001 . Painfully slow. To hit the same accuracy , points closer to 1 demand a far larger N . 3. The Classic Example: f_n(x) = x^n Plot the whole family f_n(x) = x^n on [0, 1] and let n grow. Each curve is a smooth, continuous arc — yet the graphs are pressed flat against the x -axis everywhere left of 1 , while staying pinned at height 1 at the single point x = 1 . At x = 0 : f_n(0) = 0^n = 0 for all n — already at the limit 0 . At x = 0.9 : , but only very slowly.

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