Theorem Statement:

Let $f_n, f: \Omega \to \mathbb{R}$ be functions, and assume that there exists a sequence $(x_n)$ with $x_n \in \Omega$ such that $|f_n(x_n) - f(x_n)| \not\to 0$. Then $f_n$ does not converge uniformly to $f$ on $\Omega$.

Proof:

Definitions:

1. Pointwise convergence: The sequence of functions $f_n$ converges pointwise to $f$ on $\Omega$ if for each $x \in \Omega$, we have $f_n(x) \to f(x)$ as $n \to \infty$.

2. Uniform convergence: The sequence of functions $f_n$ converges uniformly to $f$ on $\Omega$ if: $\lim_{n \to \infty} \sup_{x \in \Omega} |f_n(x) - f(x)| = 0.$ This means that the convergence of $f_n(x)$ to $f(x)$ is uniform for all $x \in \Omega$, i.e., for every $\epsilon > 0$, there exists an $N \in \mathbb{N}$ such that for all $n \geq N$ and for all $x \in \Omega$, $|f_n(x) - f(x)| < \epsilon$.

Contrapositive Statement:

We aim to prove the contrapositive of the given theorem, which is easier to handle. The contrapositive is: If $f_n$ converges uniformly to $f$ on $\Omega$, then for every sequence $(x_n)$ in $\Omega$, we have $|f_n(x_n) - f(x_n)| \to 0$.

Proof of the Contrapositive:

Assume that $f_n$ converges uniformly to $f$ on $\Omega$. This means: $\forall \epsilon > 0, \exists N \in \mathbb{N}, \forall n \geq N, \forall x \in \Omega, \quad |f_n(x) - f(x)| < \epsilon.$ Now, let $(x_n)$ be any sequence in $\Omega$. Since uniform convergence implies that $|f_n(x) - f(x)|$ is small for all $x \in \Omega$ simultaneously when $n$ is large enough, we have that for all $n \geq N$, and specifically for $x = x_n$: $|f_n(x_n) - f(x_n)| < \epsilon.$ This shows that $|f_n(x_n) - f(x_n)| \to 0$ as $n \to \infty$, because for any $\epsilon > 0$, there exists $N$ such that $|f_n(x_n) - f(x_n)| < \epsilon$ for all $n \geq N$.

Conclusion:

Thus, if $f_n$ converges uniformly to $f$ on $\Omega$, then for every sequence $(x_n)$ in $\Omega$, we have $|f_n(x_n) - f(x_n)| \to 0$. The contrapositive of this statement is exactly the given theorem, which proves that if there exists a sequence $(x_n)$ such that $|f_n(x_n) - f(x_n)| \not\to 0$, then $f_n$ does not converge uniformly to $f$ on $\Omega$.

This completes the proof.

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Would you like more details or any clarifications? Here are some related questions to explore:

1. What is the difference between pointwise and uniform convergence?
2. How do you prove uniform convergence of a sequence of functions?
3. Can you give an example where $f_n$ converges pointwise but not uniformly?
4. What are the consequences of uniform convergence for the continuity of the limit function?
5. How does Dini's theorem relate to uniform convergence?

Tip: Uniform convergence ensures that many properties, such as continuity, integration, and differentiation, pass to the limit function.