The Uniform Cauchy Criterion

From Pairwise Closeness to an Error Estimate

In “Uniform Cauchy Sequences,” uniform Cauchy behavior was defined by requiring every sufficiently late pair of functions to be close at every point of the domain. The Uniform Cauchy Criterion, stated earlier in this course, connects that pairwise condition to uniform convergence: a sequence of real-valued functions is uniformly Cauchy on its domain if and only if it converges uniformly there to a function.

The criterion is useful not only for deciding whether a uniform limit exists. It also gives a practical way to estimate the distance to a limit once the limit is known, or once pointwise convergence has been established. The key is to measure how far apart the functions in each tail of the sequence can be. These tail estimates often are easier to calculate than the limit itself.

As \(N\) increases, the collection of pairs and inputs in this supremum can only shrink. Consequently, \(D_{N+1}\leq D_N\). The uniform Cauchy condition says precisely that these tail discrepancies become arbitrarily small. When checking the definition directly, it is enough to find a bound on \(D_N\) that tends to zero. A supremum need not be attained: a bound on every pairwise difference still bounds the supremum, even if no particular pair realizes it.

The distinction between a pointwise estimate and a tail discrepancy is important. A bound that depends on \(x\) may prove that the values at each fixed input form a Cauchy sequence, but it need not give one small bound for the whole domain. To establish uniform Cauchy behavior, the bound must control all \(x\in E\) with the same choice of \(N\).

A Tail Bound Controls the Error to the Limit

The first result turns pairwise bounds into bounds against a limit. It assumes pointwise convergence, so the limit can be used in an ordinary real-variable limit at each fixed input. The resulting estimate is uniform because the original pairwise bound did not depend on the input.

Proof. Fix \(N\) with \(D_N<+\infty\), an index \(n\geq N\), and an input \(x\in E\). For every \(m\geq N\), both \(m\) and \(n\) belong to the tail defining \(D_N\). Therefore

As \(m\) tends to infinity, \(f_m(x)\) tends to \(f(x)\). The absolute value is continuous, so taking the limit in this inequality gives

This holds for every \(x\in E\), with the same right-hand side. Taking the supremum over \(x\) proves the claim. \(\square\)

In particular, if \(D_N\) tends to zero, the theorem gives uniform convergence to the pointwise limit and supplies an explicit error bound for every term in the tail. The Uniform Cauchy Criterion also guarantees a uniform limit from the uniform Cauchy condition alone; the estimate above adds quantitative information about how quickly the functions approach that limit.

Tail Bounds for Bounded Linear Combinations

A frequent source of uniformly Cauchy sequences is a finite linear combination of fixed bounded functions, with scalar coefficients that vary with \(n\). The estimates separate naturally: changes in a coefficient are multiplied by the size of its fixed function. Boundedness ensures that these multipliers can be controlled uniformly over the domain.

Proof. Write \(M=\|g\|_\infty\) and \(K=\|h\|_\infty\), both of which are finite. Fix \(\varepsilon>0\), and choose

Since \((a_n)\) and \((b_n)\) are Cauchy sequences of real numbers, there is an index \(N\) such that for \(m,n\geq N\),

For every \(x\in E\), subtract the formulas for \(u_m(x)\) and \(u_n(x)\), then apply the triangle inequality:

The second line is valid because \(|g(x)|\leq M\), \(|h(x)|\leq K\), and both coefficient differences are less than \(\delta\). The final strict inequality holds since \(M+K\geq0\) and \(M+K<M+K+1\). If \(M=K=0\), the bound is \(0<\varepsilon\), so the zero-function case is included as well. The index \(N\) does not depend on \(x\), proving that \((u_n)\) is uniformly Cauchy. \(\square\)

When Tail Discrepancies Do Not Shrink

To disprove uniform Cauchy behavior, it is enough to find a fixed positive separation that persists arbitrarily far into the sequence. The input at which that separation occurs may depend on the indices. This is closely related to the moving-input method for testing uniform convergence, but the comparison here is between two functions in a late tail.

Using the Criterion Without Confusing Its Quantifiers

A tail estimate should be checked against all three parts of the uniform Cauchy condition: the two indices must both lie far enough into the sequence, every input must be covered, and the final bound must become smaller than any prescribed positive accuracy. An estimate that applies only when one index is fixed, or that works only at each fixed input with an input-dependent index, does not establish uniform Cauchy behavior.

Conversely, once pointwise convergence is known, the Tail Discrepancy Bounds the Limit Error theorem makes a useful workflow possible. First estimate pairwise differences on a tail. Then take the supremum over the domain to obtain \(D_N\). Finally, apply the same bound to the distance from any sufficiently late term to the pointwise limit. This approach can provide a convergence rate without requiring a separate direct estimate of the limit error.

The main practical lesson is that pairwise tail estimates contain more information than a yes-or-no convergence test. They can identify the rate at which the sequence becomes uniformly close to itself, and, once a pointwise limit is available, that same rate controls the uniform distance to the limit. The next tutorial proves the Uniform Cauchy Criterion itself.