Scalar

Definition: A sequence $x_1, x_2, \ldots$ of real numbers is said to converge if there exists a real number $x$ such that for all $\epsilon > 0$, there is an integer $N$ such that $\abs{x_n-x} < \eps$ for all $n > N$. In this case, we say that $x_n$ converges to $x$, or that $x$ is the limit of $x_n$, and write $x_n \to x$ or $\lim x_n = x$ (these mean the same thing). If no such number exists, the sequence is said to diverge.*

See this math review for additional discussion and examples.

Vector

Definition: We say that the vector $\x_n$ converges to $\x$, denoted $\x_n \to \x$, if each element of $\x_n$ converges to the corresponding element of $\x$.

Theorem: $\x_n \to \x$ if and only if $\norm{\x_n -\x} \to 0$.

Proof: Let $\d_n = \x_n - \x$.

(i) Suppose $\d_n \to \zero$.

\[\begin{alignat*}{2} \norm{\d_n} \to \norm{\zero} = 0 &\hspace{4em}& \href{norm-continuity.html}{\text{Continuity of norms}} \\ \end{alignat*}\]

(ii) Suppose $\norm{\d_n} \to 0$.

\[\begin{alignat*}{2} &\exists \d: \d_n \to \d &\hspace{4em}& \href{cauchy.html}{\text{Cauchy convergence criterion}} \\ &\text{Suppose } \exists j: d_j \neq 0\\ &\qquad \norm{\d} \neq 0 && \href{norm.html}{\text{Positivity}} \\ &\qquad \text{Contradiction.} \end{alignat*}\]