IYH Let's apply the sup-norm criterion to demonstrate non-uniform convergence of f_n​(x)=x_n on [0,1]

The Theorem

The theorem states: A sequence of bounded functions (f_n) converges uniformly to a function f on a set A if and only if the sequence of sup-norms of the difference converges to zero:

lim (n→∞) ||f_n - f||_A = 0 where ||g||_A = sup { |g(x)| : x ∈ A }.

The Proof of Non-Uniform Convergence

We want to show that f_n(x) = x^n does not converge uniformly to f(x) on A = [0, 1]. According to the theorem, this is equivalent to showing that: lim (n→∞) ||f_n - f||_[0, 1] ≠ 0

Let's compute the sup-norm ||f_n - f||_[0, 1]:

||f_n - f||_[0, 1] = sup { |f_n(x) - f(x)| : x ∈ [0, 1] } = sup { |x^n - f(x)| : x ∈ [0, 1] }

We need to consider the definition of f(x):

• For x ∈ [0, 1): |x^n - f(x)| = |x^n - 0| = x^n
• For x = 1: |x^n - f(x)| = |1^n - 1| = |1 - 1| = 0

So, we are looking for: ||f_n - f||_[0, 1] = sup { {x^n : x ∈ [0, 1)} ∪ {0} } = sup { x^n : x ∈ [0, 1) }

Consider the function h(x) = x^n for a fixed n ≥ 1 on the interval [0, 1). This function is continuous and strictly increasing on [0, 1). The supremum of h(x) on [0, 1) is the limit as x approaches 1 from the left:

sup { x^n : x ∈ [0, 1) } = lim (x→1⁻) x^n = 1^n = 1

Therefore, for every n ≥ 1: ||f_n - f||_[0, 1] = 1

Now, we evaluate the limit of this sup-norm as n → ∞: lim (n→∞) ||f_n - f||_[0, 1] = lim (n→∞) 1 = 1

Conclusion: Since lim (n→∞) ||f_n - f||_[0, 1] = 1, and 1 ≠ 0, the necessary condition for uniform convergence given by the theorem (lim ||f_n - f||_A = 0) is not satisfied.

Therefore, the sequence of functions f_n(x) = x^n does not converge uniformly to its pointwise limit f(x) on the interval A = [0, 1].

IYH I had two LLMs work to explain the proof in simple language for beginners:

Simple Explanation of the Proof (for Beginners)

1. The Setup: What Are We Looking At?

We start with a sequence of functions defined on the interval [0, 1]. Each function is given by

  fₙ(x) = xⁿ

For example, f₁(x) = x, f₂(x) = x², f₃(x) = x³, and so on.

2. Pointwise Limit: What Happens at Each Point?

We examine what happens to fₙ(x) as n becomes very large.

 • For any point x between 0 and 1 (that is, for 0 ≤ x < 1):

  As n increases, the value xⁿ approaches 0. For instance, if x = 0.5 then 0.5ⁿ becomes 0.5, 0.25, 0.125, 0.0625, …, approaching 0.

 • At x = 1:

  Since 1 raised to any power remains 1, fₙ(1) = 1 for every n.

Thus, the limit “function” f(x) is defined as follows:

  f(x) = 0 for 0 ≤ x < 1,

  f(1) = 1.

3. Uniform vs. Pointwise Convergence: Why It Matters

Pointwise convergence means that, for each fixed value of x, the sequence fₙ(x) eventually gets as close as desired to f(x). However, uniform convergence is a stronger condition; it requires that the entire sequence of functions settles down to the limit function simultaneously on the whole interval. Think of it like a crowd of people: pointwise convergence is like each person eventually coming to a stop, while uniform convergence is like everyone stopping at exactly the same time.

4. Measuring the Difference: The Sup-Norm

To determine uniform convergence, we need to look at the maximum difference between fₙ(x) and the limit function f(x) over all x in [0, 1]. This maximum difference is measured using what is called the sup-norm.

For our functions, the difference at any point x is:

  |fₙ(x) − f(x)|.

For x in [0, 1), since f(x) = 0, this difference is simply xⁿ; for x = 1, the difference is |1 − 1| = 0.

5. Calculating the Sup-Norm over [0, 1]

Since xⁿ increases as x gets closer to 1 (even though it eventually decreases with large n when x < 1), the maximum difference occurs as x approaches 1 from the left. In fact, as x approaches 1, xⁿ becomes as close to 1 as desired—even though at exactly x = 1 the difference is 0. This means that the sup-norm, which is the maximum difference over the whole interval, remains equal to 1 for all n.

6. The Final Conclusion: Uniform Convergence Does Not Occur

For the functions to converge uniformly to the limit function f(x), the sup-norm difference would need to approach 0 as n increases. However, since the sup-norm is always 1, we have:

  limₙ→∞ (sup-norm of |fₙ(x) − f(x)|) = 1 ≠ 0.

Therefore, although the functions fₙ(x) converge pointwise to f(x), they do not converge uniformly.

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Summary

• A sequence of functions fₙ(x) = xⁿ is considered on the interval [0, 1].

• Pointwise, each x in the interval (except at x = 1) eventually leads fₙ(x) to 0, while fₙ(1) always equals 1.

• Uniform convergence requires the entire sequence to be close to the limit function at the same time, but here the maximum difference remains 1 for all n.

• Hence, the sequence converges pointwise but not uniformly.

This example shows that pointwise convergence doesn’t guarantee uniform convergence—a classic pitfall in calculus!