Constant Sequence in Normed Vector Space Converges

Theorem

Let $\Bbb F$ be a subfield of $\C$.

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

Let $x \in X$.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence with $x_n = x$ for all $n \in \N$.

Then:

$x_n \to x$

Proof

We have:

$\norm {x_n - x} = 0$

for all $n \in \N$.

So, for all $\epsilon > 0$, we have:

$\norm {x_n - x} < \epsilon$ for all $n \in \N$.

So:

$x_n \to x$

$\blacksquare$