Convergent Sequence Minus Limit/Proof 1

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:

$\ds \lim_{n \mathop \to \infty} \cmod {x_n - l} = 0$

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

$\blacksquare$

1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: Exercise $\S 4.29 \ (1) \ \text {(i)}$