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$f$ 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.

not sure about 2

I would appriciate any hint.

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  • $\begingroup$ 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 $\lim f(z_n)$? $\endgroup$ – Greg Martin May 25 at 20:20
  • $\begingroup$ ohh I see.. there will be two partial limits so it will not converge. what example of such sequences can I use? $\endgroup$ – Michael Sovich May 26 at 6:19
  • $\begingroup$ But the assumption is that it must converge. What does that tell you about the relationship between $A$ and $B$? $\endgroup$ – Greg Martin May 26 at 9:09

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