Sequence of functions of a sequence of real numbers

Question

Let $\{f_n(x)\}_{n \in \mathbb{N}}$ be a sequence of functions from $[0,1]$ real interval to $\mathbb{R}$ such that $$ \forall n \in \mathbb{N}: f_n \text{ is injective and continuous} $$ Let $\{x_n\}_{n \in \mathbb{N}}$ and $\{y_n\}_{n \in \mathbb{N}}$ be two sequence of $[0,1]$ such that $$ \lim_{n \to \infty} f_n(x_n) = \lim_{n \to \infty} f_n(y_n) $$ I would like to know if is it true that: $$ \lim_{n \to \infty} x_n = \lim_{n \to \infty} y_n $$

Thanks.

Answer 1

Not in general.

Let it be that every $f_n$ is prescribed by $x\mapsto \frac{x}{n}$.

If $x_n=x\neq y=y_n$ then: $$\lim_{n\to\infty} f_n(x_n)=0=\lim_{n\to\infty} f_n(y_n)$$

But: $$\lim_{n\to\infty} x_n=x\neq y=\lim_{n\to\infty} y_n$$

Answer 2

No, that's not true. There are simple examples meeting the prerequisites which converge to a constant function (look at $f_n(x) = x^n$ on $[0,1/2]$ and scale to get it on $[0,1]$) .