The set "T" mapped on all real numbers. Fairly "T" seems to be subset of set "S"

Question

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $\lim _{n \rightarrow \infty} f^{n}(x)$ exists for every $x \in \mathbb{R}$, where

\begin{aligned} f^{n}(x)=f \circ f^{n-1}(x) \text { for } n & \geq 2 . \text { Define } \\ S &=\left\{\lim _{n \rightarrow \infty} f^{n}(x): x \in \mathbb{R}\right\} \text { and } T=\{x \in \mathbb{R}: f(x)=x\} \end{aligned}

Then which of the following is necessarily true?

A. $S \subset T$

B. $T \subset S$

$\mathrm{C} . S=T$

D. None of the above

Answer 1

I'm sorry, but you have not done anything here that makes sense.

• You wrote out the definition of $f^2(x)$ and $f^3(x)$.
• You proclaimed "putting the value of $f(x) = x$" without any explanation of what you are trying to accomplish.
• Then you immediately jump to $S = \{f(x):x \in \Bbb R\}$, which does not at all follow from what you had written before.

Why?? What are you attempting to do with this?

For A) you have to show that every point of $S$ is also a point of $T$. That is, if $y$ is a real number for which $\lim_{n \to \infty} f^n(x) = y$ for some real number $x$, then $f(y) = y$. This would prove A true. Either that, or else give an example of a continuous function $f$ for which $\lim_n f^n$ always converges, and a value $y \in S$ for which $y \ne f(y)$. This would prove A false.

For B) you have to show that if every point in $T$ is also a point of $S$. That is, if $f(y) = y$, then $y = \lim_{n\to \infty} f^n(x)$ for some $x$. This would prove B true. Either that or else give an example of such a function $f$ and a value $y$ such that $f(y) = y$, but $y \ne \lim_{n\to \infty} f^n(x)$ for any $x$. This would prove B false.

C is true if and only if both of A and B are true. So if you prove both A and B, you've also proven C. If you found a counter-example to A or to B, you've found a counter-example to C.

D is true if and only if both of A and B are false (since this implies C is false as well).