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Real Vector Sequences#

Definition: Real Vector Sequence

A real vector sequence in the Euclidean space \(\mathbb{R}^n\) is a sequence of real vectors in \(\mathbb{R}^n\).

Convergence#

Theorem: Convergence via Metric

A real vector sequence \((\mathbf{x})_{k \in \mathcal{D}}\) in \(\mathbb{R}^n\) converges to the limit \(\mathbf{L} \in \mathbb{R}^n\) if and only the real sequence of the Euclidean distance between \(\mathbf{x}_k\) and \(\mathbf{L}\) converges to zero:

\[\lim_{k \to \infty} \mathbf{x}_k = \mathbf{L} \qquad \iff \qquad \lim_{k \to \infty} ||\mathbf{x}_k - \mathbf{L}|| = 0\]
Proof

TODO

Theorem: Convergence via Component Convergence

A real vector sequence \((\mathbf{x})_{k \in \mathcal{D}}\) in \(\mathbb{R}^n\) converges to the limit \(\mathbf{L} = \begin{bmatrix}L^1, \dotsc, L^n\end{bmatrix}^{\mathsf{T}} \in \mathbb{R}^n\) if and only if the real sequences \((x^1)_{k \in \mathcal{D}}, \dotsc, (x^n)_{k \in \mathcal{D}}\) of its components converge to \(L^1, \dotsc, L^n\), respectively:

\[\lim_{k \to \infty} \mathbf{x}_k = \mathbf{L} \qquad \iff \qquad \lim_{k \to \infty} {x^i}_k = L_i \text{ for all } i \in \{1, \dotsc, n\}\]
Example

Consider the real vector sequence \((\mathbf{x}_k)_{k \in \mathbb{N}}\) in \(\mathbb{R}^2\) defined as follows:

\[\mathbf{x}_k = \begin{bmatrix}1 + \frac{(-1)^k}{k} \\ \frac{3}{k^2}\end{bmatrix}\]

The real sequences \((x^1)_{k \in \mathbb{N}}\) and \((x^2)_{k \in \mathbb{N}}\) of its components are given as follows:

\[{x^1}_k = 1 + \frac{(-1)^k}{k}\]
\[{x^2}_k = \frac{3}{k^2}\]

They converge to the following limits, respectively:

\[\lim_{k \to \infty} {x^1}_k = \lim_{k \to \infty} 1 + \frac{(-1)^k}{k} = 1\]
\[\lim_{k \to \infty} {x^2}_k = \lim_{k \to \infty} \frac{3}{k^2} = 0\]

Therefore, the real vector sequence \((\mathbf{x}_k)_{k \in \mathbb{N}}\) converges to \(\begin{bmatrix}1 & 0\end{bmatrix}^{\mathsf{T}}\):

\[\lim_{k \to \infty} \mathbf{x}_k = \lim_{k \to \infty} \begin{bmatrix}1 + \frac{(-1)^k}{k} \\ \frac{3}{k^2}\end{bmatrix} = \begin{bmatrix}1 \\ 0\end{bmatrix}\]
Proof

TODO