Uniform Limit Theorem
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Theorem
Let $\struct {M, d_M}$ and $\struct {N, d_N}$ be metric spaces.
Let $\sequence {f_n}$ be a sequence of mappings from $M$ to $N$ such that:
- $(1): \quad \forall n \in \N: f_n$ is continuous at every point of $M$
- $(2): \quad \sequence {f_n}$ converges uniformly to $f$
Then:
- $f$ is continuous at every point of $M$.
Proof
Let $a \in M$.
We are given that $d_N$ is a metric on $N$.
By applying Metric Space Axiom $\text M 2$ twice:
| \(\text {(3)}: \quad\) | \(\ds \forall n \in \N, \forall x \in M: \, \) | \(\ds \map {d_N} {\map f x, \map f a}\) | \(\le\) | \(\ds \map {d_N} {\map f x, \map {f_n} x} + \map {d_N} {\map {f_n} x, \map {f_n} a} + \map {d_N} {\map {f_n} a, \map f a}\) |
Let $\epsilon \in \R_{>0}$.
Since $\sequence {f_n}$ converges uniformly to $f$:
| \(\text {(4 a)}: \quad\) | \(\ds \exists \NN \in \R_{>0}: \forall n \in \N_{>\NN}: \forall x \in M: \, \) | \(\ds \map {d_N} {\map f x, \map {f_n} x}\) | \(<\) | \(\ds \frac \epsilon 3\) | ||||||||||
| \(\text {(4 b)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \exists \NN \in \R_{>0}: \forall n \in \N_{>\NN}: \, \) | \(\ds \map {d_N} {\map f a, \map {f_n} a}\) | \(<\) | \(\ds \frac \epsilon 3\) | Universal Instantiation of $x$ |
We are given that $\forall n \in \N: f_n$ is continuous.
Hence:
| \(\text {(5)}: \quad\) | \(\ds \forall n \in \N: \exists \delta \in \R_{>0}: \forall x \in M: \, \) | \(\ds \map {d_M} {x, a}\) | \(<\) | \(\ds \delta\) | ||||||||||
| \(\ds \implies \ \ \) | \(\ds \map {d_N} {\map {f_n} x, \map {f_n} a}\) | \(<\) | \(\ds \frac \epsilon 3\) |
Combining $(3)$, $\text {(4 a)}$, $\text {(4 b)}$ and $(5)$:
| \(\ds \exists \NN \in \R_{>0}: \forall n \in \N_{>\NN}: \exists \delta \in \R_{>0}: \forall x \in M: \, \) | \(\ds \map {d_M} {x, a}\) | \(<\) | \(\ds \delta\) | |||||||||||
| \(\ds \implies \ \ \) | \(\ds \map {d_N} {\map f x, \map f a}\) | \(<\) | \(\ds \frac \epsilon 3 + \frac \epsilon 3 + \frac \epsilon 3\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \epsilon\) |
As $a$ and $\epsilon$ are arbitrary, it follows by Universal Instantiation of $n$ that:
| \(\ds \forall a \in M: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in M: \, \) | \(\ds \map {d_M} {x, a}\) | \(<\) | \(\ds \delta\) | |||||||||||
| \(\ds \implies \ \ \) | \(\ds \map {d_N} {\map f x, \map f a}\) | \(<\) | \(\ds \epsilon\) |
Hence, $f$ is continuous at every point of $M$.
$\blacksquare$
Sources
- 2005: G. Auliac and J.-Y. Caby: Mathématiques, Topologie et Analyse: $\S 5.2$, Theorem $5.10$