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