LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (z-ai/glm-5.2)verified 2026-07-29 (claude-fable-5)

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If for every $ε > 0$ some continuous $g : X \to R$ satisfies $∣ f (x) - g (x) ∣ < ε$ for all $x$ , then $f$ is continuous; in particular a uniformly convergent series of continuous real functions has a continuous sum

Statement

Let $(X, T)$ be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison) and let $f : X \to R$ . If for every real $ε > 0$ there is a continuous $g_{ε} : X \to R$ (Continuity of a map of topological spaces at a point and globally) with

$∣ f (x) - g_{ε} (x) ∣ < ε for every x \in X,$

then $f$ is continuous.

In particular, if $(g_{n})_{n \in N}$ are continuous real-valued functions on $X$ and $(M_{n})_{n \in N}$ are nonnegative reals with $∣ g_{n} (x) ∣ \leq M_{n}$ for every $x \in X$ and every $n$ , and the series $\sum M_{n}$ converges (Series, partial sums, convergence and the sum, divergence, and the tail series), then for every $x \in X$ the series $\sum g_{n} (x)$ converges, and

$F (x) : = \sum_{n = 0}^{\infty} g_{n} (x)$

defines a continuous function $F$ on $X$ .

Facts & Assumptions

Given: A topological space $(X, T)$ and $f : X \to R$ such that for every real $ε > 0$ there is a continuous $g_{ε} : X \to R$ with $∣ f (x) - g_{ε} (x) ∣ < ε$ for every $x \in X$ ; and, for the second clause, continuous $g_{n} : X \to R$ and nonnegative reals $M_{n}$ , $n \in N$ , with $∣ g_{n} (x) ∣ \leq M_{n}$ for every $x \in X, n \in N$ , and $\sum M_{n}$ convergent.

[A1]

The main hypothesis: for every real $ε > 0$ there is continuous $g_{ε}$ with $∣ f (x) - g_{ε} (x) ∣ < ε$ for all $x \in X$ .

[L1]

$f$ is continuous at $x_{0}$ iff for every open $V \subseteq R$ with $f (x_{0}) \in V$ there is open $U \subseteq X$ with $x_{0} \in U$ and $f [U] \subseteq V$ (Continuity of a map of topological spaces at a point and globally).

[L2]

Preimages of open sets under a continuous map are open (For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and $f (\overline{A}) \subseteq \overline{f (A)}$ , clause (b)).

[L3]

The bounded open intervals $(a - r, a + r)$ , $r > 0$ , are a basis for the usual topology of $R$ , so for open $V ∋ y_{0}$ there is real $r > 0$ with $(y_{0} - r, y_{0} + r) \subseteq V$ (The absolute value makes $R$ a metric space: $d (x, y) = ∣ x - y ∣$ is a metric, its open balls are the intervals $(x - r, x + r)$ , and it is unbounded, claim 2 and 3, Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Basis and subbasis for a topology, and the topology generated by a family of sets).

[L4]

Triangle inequality: $∣ u + v ∣ \leq ∣ u ∣ + ∣ v ∣$ , hence $∣ u - w ∣ \leq ∣ u - v ∣ + ∣ v - w ∣$ for reals $u, v, w$ (The triangle inequality).

[L5]

Absolute value: $∣ u ∣ < c$ iff $- c < u < c$ , for real $c > 0$ ; and $- c \leq u \leq c$ iff $∣ u ∣ \leq c$ , for real $c \geq 0$ (Basic properties of the absolute value).

[L6]

Finite triangle inequality along a finite index set, iterating [L4]: $∣ \sum_{k} u_{k} ∣ \leq \sum_{k} ∣ u_{k} ∣$ (Basic properties of the absolute value, Ordered field).

[L7]

Comparison and absolute convergence: if $0 \leq a_{k} \leq b_{k}$ eventually and $\sum b_{k}$ converges then $\sum a_{k}$ converges (If $0 \leq a_{k} \leq b_{k}$ eventually, convergence of $\sum b_{k}$ gives convergence of $\sum a_{k}$ , and divergence of $\sum a_{k}$ gives divergence of $\sum b_{k}$ ); if $\sum ∣ a_{k} ∣$ converges then $\sum a_{k}$ converges (If $\sum ∣ a_{k} ∣$ converges then $\sum a_{k}$ converges).

[L8]

For a series of nonnegative terms, the partial sums are nondecreasing, bounded above by the sum when the series converges, and converge to the sum (A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum, Series, partial sums, convergence and the sum, divergence, and the tail series).

[L9]

Limits in $R$ preserve non-strict order: if $a_{k} \to a$ and $a_{k} \leq c$ for all $k$ beyond some index, then $a \leq c$ (Sequence basics in an arbitrary ordered field: limits are unique, limits preserve non-strict inequalities, convergent sequences are Cauchy, Cauchy sequences are bounded, and a Cauchy sequence with a convergent subsequence converges).

Proof

technique · constructive

1.1

Fix $x_{0} \in X$ and an open $V \subseteq R$ with $f (x_{0}) \in V$ ; by [L3] fix a real $r > 0$ with $(f (x_{0}) - r, f (x_{0}) + r) \subseteq V$ .

givenL3choose

1.2

Let $g, h : X \to R$ be continuous, let $x_{1} \in X$ and let real $η > 0$ ; arguing directly from continuity of $g$ and of $h$ at $x_{1}$ (via [L1] and [L2]) separately, fix open $U_{1}, U_{2} ∋ x_{1}$ with $∣ g (x) - g (x_{1}) ∣ < η / 2$ on $U_{1}$ and $∣ h (x) - h (x_{1}) ∣ < η / 2$ on $U_{2}$ .

givenL1L2choose

1.3

Fix $x \in X$ . The real sequence $(g_{n} (x))_{n \in N}$ satisfies $0 \leq ∣ g_{n} (x) ∣ \leq M_{n}$ for every $n$ , and $\sum M_{n}$ converges by hypothesis, so [L7] gives that $\sum ∣ g_{n} (x) ∣$ converges, and hence $\sum g_{n} (x)$ converges; define $F (x) : = \sum_{n = 0}^{\infty} g_{n} (x)$ and $s_{N} (x) : = \sum_{n < N} g_{n} (x)$ , so $s_{N} (x) \to F (x)$ as $N \to \infty$ .

givenL7construct

1.4

Write $σ_{N} : = \sum_{n < N} M_{n}$ and $S : = \sum_{n = 0}^{\infty} M_{n}$ ; since $M_{n} \geq 0$ for every $n$ , [L8] gives that $(σ_{N})$ is nondecreasing with $σ_{N} \leq S$ for every $N$ , and $σ_{N} \to S$ . So $S - σ_{N} \geq 0$ for every $N$ and $S - σ_{N} \to 0$ ; given a real $ε > 0$ , fix $N \in N$ with $S - σ_{N} < ε$ .

givenL8choose

2.1

By [A1] applied with $ε : = r / 3 > 0$ , fix a continuous $g : X \to R$ with $∣ f (x) - g (x) ∣ < r / 3$ for every $x \in X$ .

step 1.1A1choose

2.2

$U_{1} \cap U_{2}$ is open, contains $x_{1}$ , and for $x \in U_{1} \cap U_{2}$ : $∣ (g + h) (x) - (g + h) (x_{1}) ∣ \leq ∣ g (x) - g (x_{1}) ∣ + ∣ h (x) - h (x_{1}) ∣ < η$ by [L4].

step 1.2L4algebra

2.3

For every $x \in X$ and every $K > N$ : $∣ s_{K} (x) - s_{N} (x) ∣ = ∣ \sum_{N \leq n < K} g_{n} (x) ∣ \leq \sum_{N \leq n < K} ∣ g_{n} (x) ∣ \leq \sum_{N \leq n < K} M_{n} = σ_{K} - σ_{N} \leq S - σ_{N}$ , by [L6], the hypothesis $∣ g_{n} (x) ∣ \leq M_{n}$ , and $σ_{K} \leq S$ from step 1.4.

step 1.4step 1.3L6algebra

3.1

$U : = g^{- 1} [(g (x_{0}) - r / 3, g (x_{0}) + r / 3)]$ is open by [L2], since $g$ is continuous by step 2.1, and $x_{0} \in U$ , since $∣ g (x_{0}) - g (x_{0}) ∣ = 0 < r / 3$ .

step 2.1L2

3.2

Since $x_{1} \in X$ and real $η > 0$ were arbitrary, $g + h$ is continuous on $X$ ; iterating this over finitely many further sums, any finite sum $g_{0} + \dots + g_{N - 1}$ of continuous real-valued functions on $X$ is continuous, for every $N \geq 1$ , with the case $N = 0$ (the zero function) continuous as a constant.

step 2.2

3.3

By step 2.3, $∣ s_{K} (x) - s_{N} (x) ∣ \leq S - σ_{N}$ for every $K > N$ ; as $K \to \infty$ , $s_{K} (x) \to F (x)$ by step 1.3, so [L9] applied to the two non-strict bounds $- (S - σ_{N}) \leq s_{K} (x) - s_{N} (x) \leq S - σ_{N}$ (equivalent to step 2.3 by [L5]) gives $- (S - σ_{N}) \leq F (x) - s_{N} (x) \leq S - σ_{N}$ , that is $∣ F (x) - s_{N} (x) ∣ \leq S - σ_{N} < ε$ by [L5] and step 1.4, for every $x \in X$ , with $N$ independent of $x$ .

step 2.3step 1.4step 1.3L5L9

4.1

For $x \in U$ : $∣ f (x) - f (x_{0}) ∣ \leq ∣ f (x) - g (x) ∣ + ∣ g (x) - g (x_{0}) ∣ + ∣ g (x_{0}) - f (x_{0}) ∣ < r / 3 + r / 3 + r / 3 = r$ , by [L4] (twice), step 2.1 (the first and third terms) and the defining property of $U$ (step 3.1, the middle term).

step 2.1step 3.1L4algebra

4.2

For $N \in N$ , $s_{N} = g_{0} + \dots + g_{N - 1}$ is a finite sum of continuous functions, hence continuous on $X$ , by step 3.2.

step 3.2

5.1

By step 4.1, $f (x) \in (f (x_{0}) - r, f (x_{0}) + r) \subseteq V$ for every $x \in U$ (step 1.1), so $f [U] \subseteq V$ ; with $U$ open and $x_{0} \in U$ (step 3.1), and $V$ an arbitrary open set containing $f (x_{0})$ (step 1.1), $f$ is continuous at $x_{0}$ by [L1].

step 4.1step 3.1step 1.1L1

6.1

Since $x_{0} \in X$ was arbitrary, $f$ is continuous on $X$ ; this proves the main clause.

step 5.1

7.1

Since $s_{N}$ is continuous by step 4.2 and real $ε > 0$ was arbitrary, the hypothesis of the main clause (steps 1.1–6.1) is met by $F$ , taking $g_{ε} : = s_{N}$ ; hence $F$ is continuous on $X$ . This, with step 1.3, proves the second clause.

step 3.3step 4.2step 6.1discharge-construct∎

Remarks

The $ε / 3$ split is the whole mechanism, and it is exactly the triangle inequality read three ways: once to compare $f$ with an approximant, once to use continuity of that approximant, and once to compare back. Nothing about $X$ is used beyond the definition of continuity; the hypothesis never mentions a metric on $X$ , only on the common target $R$ .
The second clause is the Weierstrass $M$ -test, stated only as far as this page needs it. It is not stated for a general metric or normed target, and it produces no rate of convergence beyond what step 1.4 already gives: a single $N$ , working uniformly in $x$ , for every tolerance $ε$ .
No choice principle beyond what a single real number requires is used anywhere above. Steps 1.1, 2.1 and 1.4 each fix one witness from a nonempty set of reals or a single continuous function, and no step selects simultaneously from an infinite family.

Depends on

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69 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Uniform convergence (Wikipedia) (standard reference, not scraped)
Weierstrass M-test (Wikipedia) (standard reference, not scraped)
J. Munkres, Topology, 2nd ed., §21 (standard reference, not scraped)

If for every ε>0ε>0 some continuous g:X→Rg:X→R satisfies ∣f(x)−g(x)∣<ε∣f(x)−g(x)∣<ε for all xx, then ff is continuous; in particular a uniformly convergent series of continuous real functions has a continuous sum