# functions convergence

$$f$$ is bounded function defined around $$x_0$$. For every monotonic sequence $$x_n \rightarrow x_0$$, $$f(x_n)$$ is convergent.

prove/disprove :

1. all $$f(x_n)$$ sequences converge to the same limit when $$x_n\rightarrow x_0$$ is monotonic increasing func.

2. one sided limits of $$f$$ exist at $$x_0$$.

I think 1 is not correct so I am trying to find 2 monotonic sequences $$x_n,y_n$$ that converge to $$x_0$$ s.t $$\lim f(x_n)$$ Differ from $$\lim f(y_n)$$, but no success so far.

• You might reconsider your intuition about #1. Given $x_n$ and $y_n$ and the associated limits $f(x_n)\to A$ and $f(y_n)\to B$, let $\{z_n\}$ be the union of $\{x_n\}$ and $\{y_n\}$ (sorted to be monotonic); what can you say about $\lim f(z_n)$? – Greg Martin May 25 at 20:20
• But the assumption is that it must converge. What does that tell you about the relationship between $A$ and $B$? – Greg Martin May 26 at 9:09