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“It Looks True” Versus “It Is Proved”

The single deepest habit of analysis: distrusting the obvious until it is established by [[proof|proof]]. We see a claim that looks certain and is false, learn why a picture is evidence but never a verdict, and meet the one move that saves you from a thousand wrong theorems — the [[counterexample|counterexample]].

A claim that looks true and isn't

Here is a statement that feels obviously true: “if each function in a sequence is continuous, and the sequence converges to a limit function, then the limit is continuous too.” Continuity should survive a limiting process — what could go wrong? Yet it is false. One explicit example demolishes it, and watching that demolition teaches the entire ethic of rigor.

Claim:  f_n continuous for every n,  and  f_n -> f,  imply  f is continuous.

Counterexample on [0, 1].   Let  f_n(x) = x^n.

  Each f_n is a polynomial -> continuous everywhere. Good.

  Find the limit f(x) = lim_{n->inf} f_n(x), point by point:
    if 0 <= x < 1 :  x^n -> 0       (a number below 1, raised to ever
                                     higher powers, shrinks to 0)
    if x = 1     :  1^n = 1 -> 1

  So the limit function is
    f(x) = 0   for  0 <= x < 1
    f(1) = 1

  Is f continuous at x = 1?
    Approach 1 from the left: f(x) = 0, so the left-hand limit is 0.
    But f(1) = 1.   0  is NOT  1.
    -> f has a JUMP at x = 1. It is DISCONTINUOUS.

Every f_n was continuous; the limit f is not. The claim is FALSE.
Each f_n(x) = x^n is a smooth, continuous curve, yet their pointwise limit jumps. The plausible claim is dead — and we learn that continuity does NOT automatically pass to a limit.

Notice what the example did. The claim was a “for all” statement — *for every* such sequence, the limit is continuous. To destroy a “for all” claim you need just one case where it fails. That single case is a counterexample, and it is the most economical weapon in mathematics: one example outranks any amount of plausibility.

Why a picture is evidence, not proof

A graph shows finitely many pixels at finite resolution. It can suggest a pattern, and good mathematicians lean on pictures constantly to find what to prove. But a picture cannot rule out behaviour finer than a pixel, nor behaviour out at infinity, nor behaviour at a single exceptional point. The x^n example hides its jump in an infinitely thin sliver near x = 1 that no finite plot resolves.

What a proof actually has to do

A proof is not rhetoric or a high level of confidence; it is a finite chain of steps, each one following from definitions and earlier-proved facts, that forces the conclusion in *every* permitted case at once. Plausibility is a feeling about one example; a proof is a guarantee about all of them. That is why one counterexample beats a thousand confirming examples but a thousand confirming examples never make a proof.

There is a hopeful flip side. Our broken claim was almost right; the cure is to add the missing hypothesis. Strengthen “converges” to *uniform* convergence and the limit of continuous functions really is continuous (the uniform limit theorem). Analysis does not just say no — it finds the exact extra condition that makes the theorem true, which is what necessary and sufficient conditions are about.