What are some examples of convergent sequences that do not converge uniformly?

Sequences and their convergence are fundamental concepts in mathematical analysis. While a sequence can converge to a limit, the convergence can sometimes be uniform or non-uniform. In this article, we delve into an example of a convergent sequence that does not converge uniformly, providing a detailed explanation and an illustrative example using the sequence fn : xn in the closed unit interval [0, 1].

Convergent Sequence in the Unit Interval [0, 1]

Let's consider the sequence defined by fn(x) xn, where n 1, 2, .... This sequence is defined over the closed unit interval [0, 1].

Pointwise Convergence

Firstly, we explore the pointwise convergence of this sequence. As n approaches infinity, xn approaches 0 for all x in (0, 1]. At x 0, the sequence converges to 0 trivially. At x 1, the sequence converges to 1. Therefore, the pointwise limit function f(x) is given by:

[f(x) begin{cases} 0, text{if } x in [0, 1) 1, text{if } x 1 end{cases}]

Uniform Convergence Failure

Now, let's investigate the uniform convergence of this sequence. For a sequence to converge uniformly to a limit function f(x), the convergence must be uniform across the entire interval. In other words, given any epsilon; 0, there exists an N such that |fn(x) - f(x)| epsilon; for all x in the interval and for all n N.

To see why this sequence does not converge uniformly, consider a value x in the interval (0, 1), for example, x frac{1}{2}. The sequence xn at x frac{1}{2} is left(frac{1}{2}right)^1, left(frac{1}{2}right)^2, left(frac{1}{2}right)^3, ldots, which converges to 0. However, this convergence is not uniform because the convergence is indefinitely slow for x close to 1.

Illustrative Example

Consider the specific example where x frac{1}{2}. The sequence is as follows:

[frac{1}{2}, left(frac{1}{2}right)^2, left(frac{1}{2}right)^3, ldots, left(frac{1}{2}right)^{100}, ldots]

Each term in the sequence is:

[left(frac{1}{2}right)^1 frac{1}{2}] [left(frac{1}{2}right)^2 frac{1}{4}] [left(frac{1}{2}right)^3 frac{1}{8}] [ldots] [left(frac{1}{2}right)^{100} frac{1}{2^{100}}]

The sequence xn becomes indefinitely small as n increases, but the rate of decrease is not uniform across the interval. This slowness in convergence near x 1 prevents uniform convergence.

Implications of Non-uniform Convergence

The failure of uniform convergence of xn implies that the limit function is not continuous. To understand why, recall that the uniform limit of a sequence of continuous functions is itself continuous if and only if the sequence converges uniformly. Here, the sequence of functions fn(x) does not converge uniformly to the limit function f(x).

Hence, the limit function f(x) is discontinuous at x 1. This is a key property of sequences that converge non-uniformly.

Conclusion

In this article, we have explored a classic example of a sequence fn(x) xn in the closed unit interval [0, 1] that converges pointwise but not uniformly. The failure of uniform convergence is demonstrated through the example of x frac{1}{2}, where the sequence approaches 0 but very slowly near 1. This example highlights the importance of the distinction between pointwise and uniform convergence in mathematical analysis.