Sequences in Finite-Dimensional Spaces

Sequences Through Coordinates

In a finite-dimensional space, a basis represents every vector by finitely many real coordinates. This makes sequences easier to analyze: convergence of vectors can be tested coordinate by coordinate, and bounded sequences can be handled by applying the Bolzano–Weierstrass Theorem to those coordinates. The previous tutorial proved that all norms on a finite-dimensional space are equivalent. We now use that result to study sequences without requiring a special formula for the norm.

Fix a real vector space \(V\) of dimension \(d\geq1\), a basis \(e_1,\ldots,e_d\), and a norm \(N\) on \(V\). Write each vector \(x\in V\) uniquely as \(x=\sum_{i=1}^d a_i e_i\), and define its coordinate Euclidean norm by

The Proposition (A Norm Is Comparable to a Coordinate Norm) from the previous tutorial gives constants \(m,C>0\) such that

These two inequalities connect the norm used in the problem to the coordinates supplied by the basis. In particular, they allow us to characterize convergence exactly.

Proof. Suppose first that \(x_n\to x\) in \(N\). The lower comparison bound gives

For each coordinate, \(|a_{n,i}-a_i|\leq |x_n-x|_2\), since the square of the Euclidean norm is the sum of the squares of all coordinate differences. Therefore \(a_{n,i}\to a_i\) for every \(i\).

Conversely, suppose that every coordinate sequence converges to its corresponding coordinate. Given \(\varepsilon>0\), for each \(i\) there is an index \(n_i\) such that \(|a_{n,i}-a_i|<\varepsilon/(C\sqrt d)\) whenever \(n\geq n_i\). Since there are only finitely many coordinates, \(n_0=\max\{n_1,\ldots,n_d\}\) is defined. For \(n\geq n_0\), the upper comparison bound yields

Thus \(x_n\to x\) in \(N\). \(\square\)

The proof uses finite dimension in an important way: we take the maximum of finitely many coordinate thresholds. Coordinatewise convergence in an infinite list of coordinates does not automatically provide one threshold that works for every coordinate at once. Here the number of coordinates is fixed and finite.

Bounded Sequences Have Convergent Subsequences

A sequence \((x_n)\) in a normed space is bounded if there is a finite \(M\geq0\) such that \(N(x_n)\leq M\) for every \(n\). In finite dimensions, this condition bounds every coordinate. Indeed, the lower comparison bound gives \(m|x_n|_2\leq M\), and each coordinate has absolute value at most \(|x_n|_2\). The Bolzano–Weierstrass Theorem for real sequences can then be applied repeatedly, one coordinate at a time.

Proof. If \(V=\{0\}\), every term is zero, so the sequence itself converges. Otherwise, let \(d\geq1\), choose a basis, and write \(x_n=\sum_{i=1}^d a_{n,i}e_i\). Let \(m>0\) be the lower comparison constant for \(N\) and the coordinate Euclidean norm. If \(N(x_n)\leq M\) for every \(n\), then

Thus the first coordinate sequence is bounded in \(\mathbb{R}\). By the Bolzano–Weierstrass Theorem, it has a convergent subsequence. Denote the corresponding indices by \(n^{(1)}_1<n^{(1)}_2<\cdots\). Along these indices, the second coordinate remains a bounded real sequence, so it has a convergent subsequence. This gives indices \(n^{(2)}_1<n^{(2)}_2<\cdots\), each drawn from the first subsequence, along which both the first and second coordinates converge. Continue in this way through the finite list of \(d\) coordinates. After the \(d\)-th extraction, there are increasing indices \(k_1<k_2<\cdots\) such that every coordinate of \(x_{k_j}\) converges as \(j\to\infty\).

Let \(a_i\) be the limit of the \(i\)-th coordinate along this final subsequence, and set \(x=\sum_{i=1}^d a_i e_i\in V\). The Coordinate Characterization of Norm Convergence now gives \(x_{k_j}\to x\) in \(N\). Hence the original bounded sequence has a convergent subsequence. \(\square\)

The repeated extraction is a finite process, not an infinite one: one extraction is made for each of the \(d\) coordinates. A subsequence of a convergent real sequence still converges to the same limit, so later extractions do not undo convergence already obtained for earlier coordinates.

What These Sequence Results Do—and Do Not—Say

The bounded-subsequence theorem is a central compactness principle. In finite dimensions, boundedness prevents coordinates from escaping to infinity, while successive applications of the real Bolzano–Weierstrass Theorem produce a subsequence whose coordinates all settle to limits. The coordinate characterization then turns those scalar limits into convergence in the norm on \(V\).

The exact numerical norm is not essential. The Sequential and Boundedness Invariance theorem from the previous tutorial states that equivalent norms give the same convergent sequences and bounded sequences. Alternatively, the comparison constants above show directly why the coordinate argument works for any norm on \(V\). A bound in \(N\) bounds the coordinates, and coordinate convergence implies convergence in \(N\).

Do not confuse boundedness with convergence. The matrix example is bounded but does not converge; its different subsequences have different limits. Nor does the theorem say that every sequence has a convergent subsequence: boundedness is a necessary hypothesis. For example, the sequence \(x_n=ne_1\), where \(e_1\neq0\), is unbounded because \(N(x_n)=nN(e_1)\to\infty\).

The theorem also identifies the role of finite dimension. There are only finitely many coordinates to control, so successive extraction ends after finitely many steps. In an infinite-dimensional space, bounded sequences need not have convergent subsequences. For instance, in the space of square-summable sequences with its usual norm, the unit vectors \(u_n\) satisfy \(\|u_n-u_k\|_2=\sqrt2\) whenever \(n\neq k\). No subsequence can converge, since every convergent sequence is Cauchy.