Is it true that $\lim\limits_{n\to\infty}{f(x_{n})}=\lim\limits_{n\to\infty}{f(y_{n})}$ Let $f: \mathbb{R}\rightarrow \mathbb{R}$.
If for all sequence $x_n\rightarrow 0$, the sequence ${f(x_n)}$ is converges,
can i say that it's true that for $x_n,y_n\rightarrow 0$, i will get:
$\lim\limits_{n\to\infty}{f(x_{n})}=\lim\limits_{n\to\infty}{f(y_{n})}$ ?
Thanks.

Yes. If you would have two sequences $(x_n)_{n=1}^{\infty}$  and $(y_n)_{n=1}^{\infty}$ that converge to zero and for which $\lim_{n \to \infty} f(x_n) = L_1$ and $\lim_{n \to \infty} f(y_n) = L_2$ and $L_1 \neq L_2$ then for the sequence
$$ z_n := \begin{cases} x_n & n \textrm{ is even} \\ y_n & n \textrm{ is odd} \end{cases} $$
you would have that $z_n \to 0$ but $f(z_n)$ would have two subsequences $f(z_{2n})$ and $f(z_{2n+1})$ that converge to two different limits and so $f(z_n)$ won't have a limit.
