Modulus of Limit/Normed Vector Space

Theorem
Let $\struct {X, \norm { \, \cdot \, } }$ be a normed vector space.

Let $\sequence {x_n}$ be a convergent sequence in $R$ to the limit $x$.

That is, let $\displaystyle \lim_{n \mathop \to \infty} x_n = x$.

Then
 * $\displaystyle \lim_{n \mathop \to \infty} \norm {x_n} = \norm x$

where $\sequence { \norm {x_n} }$ is a real sequence.

Proof
By the Reverse Triangle Inequality, we have:
 * $\cmod {\norm {x_n} - \norm x} \le \norm {x_n - x}$

Hence by the Squeeze Theorem and Convergent Sequence Minus Limit, $\norm {x_n} \to \norm x$ as $n \to \infty$.