Constant Sequence in Normed Vector Space Converges
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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:
- $x_n \to x$
$\blacksquare$