Convergence and completeness

A sequence converges to $x$ when every positive tolerance eventually contains all its remaining terms around $x$. The index at which this happens may depend on the tolerance. A Cauchy sequence instead compares late terms with one another, without knowing the target. Every convergent sequence is Cauchy by the triangle inequality. The reverse implication needs a property of the ambient space: completeness. A sequence can behave perfectly while the space is missing its destination.

Rational decimal approximations to $\sqrt{2}$ are Cauchy in the usual absolute-value distance but have no rational limit. Moving from $\mathbb{Q}$ to $\mathbb{R}$ repairs that particular gap. In a complete normed vector space, called a Banach space, every norm-Cauchy sequence has a limit in the space. Completeness is also part of the definition of a Hilbert space. It is a structural guarantee about all Cauchy sequences, not a claim that every sequence converges.

For functions, pointwise convergence allows a different waiting time at every input. Uniform convergence demands a single waiting time that works throughout the domain. On $[0,1]$, the continuous functions $f_n(x)=x^n$ approach zero at every $x<1$ but remain one at $x=1$. The limiting function jumps. This cannot happen for a uniform limit of continuous functions on that domain. The narrow boundary region near one keeps defeating any proposed uniform error tolerance.

Uniform convergence on a finite interval lets us pass a limit through its integral, since the integral error is bounded by interval length times the uniform error. Passing limits through derivatives needs additional hypotheses; uniform convergence of the functions alone is insufficient. For example, $\sin(nx)/n$ converges uniformly to zero while its derivatives $\cos(nx)$ do not converge pointwise everywhere. A limit theorem is permission for a particular operation under stated assumptions, not a general license to exchange symbols.

Suppose a map $T$ takes a nonempty complete metric space into itself and shrinks every distance by a fixed factor $q<1$. Iterating $x_{n+1}=T(x_n)$ makes consecutive steps decrease geometrically. Adding the remaining geometric tail proves that the iterates are Cauchy. Completeness supplies a limit, continuity of the contraction makes that limit a fixed point, and the same distance inequality proves uniqueness. Each assumption performs a different job in the argument.

This theorem clarifies why a small update is meaningful only with a quantitative model. Consecutive differences can approach zero even while a sequence drifts without bound, as partial sums of the harmonic series demonstrate. A proven contraction factor turns the last step into a bound on all future steps. Many practical optimization methods are not global contractions, especially around saddle points or across changing data. In those cases a small update is diagnostic evidence, not this theorem's certificate.