Limit of Image of Sequence

Theorem
Let $$f$$ be a real function which is continuous on the interval $$\mathbb{I}$$.

Let $$\left \langle {x_n} \right \rangle$$ be a sequence of points in $$\mathbb{I}$$ such that $$\lim_{n \to \infty} x_n = \xi$$, where $$\xi \in \mathbb{I}$$.

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

Alternatively, this can be put as $$\lim_{n \to \infty} f \left({x_n}\right) = f \left({\lim_{n \to \infty} x_n}\right)$$.

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

Proof
From Limit of Function by Convergent Sequences, we have:

$$\lim_{x \to \xi} f \left({x}\right) = l$$ iff, for each sequence $$\left \langle {x_n} \right \rangle$$ of points of $$\left({a \, . \, . \, b}\right)$$ such that $$\forall n \in \mathbb{N}^*: x_n \ne \xi$$ and $$\lim_{n \to \infty} x_n = \xi$$, it is true that $$\lim_{n \to \infty} f \left({x_n}\right) = l$$.

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