Convergent Sequence Minus Limit

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Theorem

Let $X$ be one of the standard number fields $\Q, \R, \C$.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $X$ which converges to $l$.

That is:

$\displaystyle \lim_{n \mathop \to \infty} x_n = l$


Then:

$\displaystyle \lim_{n \mathop \to \infty} \left|{x_n - l}\right| = 0$


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 $\displaystyle \lim_{n \mathop \to \infty} x_n = l$.

$\blacksquare$


Proof 2

We note that all of $\Q, \R, \C$ can be considered as metric spaces.

Then under the usual metric:

$d \left({x_n, l}\right) = \left|{x_n - l}\right|$.

The result follows from the definition of metric: $d \left({x_n, l}\right) = 0 \iff x_n = l$.

$\blacksquare$