Convergent Sequence Minus Limit
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Theorem
Let $X$ be one of the standard number fields $\Q, \R, \C$.
Let $\sequence {x_n}$ be a sequence in $X$ which converges to $l$.
That is:
- $\ds \lim_{n \mathop \to \infty} x_n = l$
Then:
- $\ds \lim_{n \mathop \to \infty} \cmod {x_n - l} = 0$
Proof 1
Let $\epsilon > 0$.
We need to show that there exists $N$ such that:
- $\forall n > N: \size {\paren {\size {x_n - l} - 0} } < \epsilon$
But:
- $\size {\paren {\size {x_n - l} - 0} } = \size {x_n - l}$
So what needs to be shown is just:
- $x_n \to l$ as $n \to \infty$
which is the definition of $\ds \lim_{n \mathop \to \infty} x_n = l$.
$\blacksquare$
Proof 2
We note that all of $\Q, \R, \C$ can be considered as metric spaces.
Then under the usual metric:
- $\map d {x_n, l} = \cmod {x_n - l}$.
The result follows from the definition of metric:
- $\map d {x_n, l} = 0 \iff x_n = l$.
$\blacksquare$