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 $\ds \lim_{n \mathop \to \infty} x_n = x$.

Then:

$\ds \lim_{n \mathop \to \infty} \norm {x_n} = \norm x$

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

Proof

By the Reverse Triangle Inequality:

$\cmod {\norm {x_n} - \norm x} \le \norm {x_n - x}$

Hence by the Squeeze Theorem and Convergent Sequence Minus Limit:

$\norm {x_n} \to \norm x$

as $n \to \infty$.

$\blacksquare$

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.4$: Normed and Banach spaces. Sequences in a normed space; Banach spaces