Sequence in Normed Vector Space Convergent to Limit iff Norm of Sequence minus Limit is Null Sequence

Theorem

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

Let $x \in X$.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$.

Then $\sequence {x_n}_{n \mathop \in \N}$ converges to $x$ if and only if:

$\norm {x_n - x} \to 0$

Proof

From the definition of a convergent sequence in a normed vector space, we have that:

$x_n$ converges to $x$

if and only if:

for each $\epsilon > 0$ there exists $N \in \N$ such that $\norm {x_n - x} < \epsilon$.

From the definition of a convergent real sequence, we have that:

$\norm {x_n - x} \to 0$

if and only if:

for each $\epsilon > 0$ there exists $N \in \N$ such that $\size {\norm {x_n - x} - 0} < \epsilon$.

Since the norm is non-negative, we have that:

$\norm {x_n - x} \to 0$

if and only if:

for each $\epsilon > 0$ there exists $N \in \N$ such that $\norm {x_n - x} < \epsilon$.

We can therefore immediately deduce the result.

$\blacksquare$