Limit of Image of Sequence/Real Number Line

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Theorem

Let $f$ be a real function which is continuous on the interval $\Bbb I$.

Let $\left \langle {x_n} \right \rangle$ be a sequence of points in $\Bbb I$ such that:

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

where:

$(1): \quad \xi \in \Bbb I$
$(2): \quad \displaystyle \lim_{n \mathop \to \infty} x_n$ denotes the limit of $x_n$.


Then:

$\displaystyle \lim_{n \mathop \to \infty} f \left({x_n}\right) = f \left({\xi}\right)$

That is:

$\displaystyle \lim_{n \mathop \to \infty} f \left({x_n}\right) = f \left({\lim_{n \mathop \to \infty} x_n}\right)$


That is, for a continuous function, the limit and function symbols commute.


Proof

From Limit of Function by Convergent Sequences, we have:

$\displaystyle \lim_{x \mathop \to \xi} f \left({x}\right) = l$

if and only if:

for each sequence $\left \langle {x_n} \right \rangle$ of points of $\left({a \,.\,.\, b}\right)$ such that:
$\forall n \in \N_{>0}: x_n \ne \xi$
and:
$\displaystyle \lim_{n \mathop \to \infty} x_n = \xi$
it is true that:
$\displaystyle \lim_{n \mathop \to \infty} f \left({x_n}\right) = l$

The result follows directly from this and the definition of continuity on an interval.

$\blacksquare$


Sources