Connectedness of a set consisting of limit points Let $f:\mathbb{R} \to \mathbb{R}$ be continuous and
$A=\{y\in\mathbb{R}:y=\lim_\limits{n\to\infty}f(x_n),x_n\to \infty\}$
What are the properties of $A$  which are always true:
1.The set is compact
2.The set is a singleton 
3.The set is connected
I think it is connected  always because it is a set of limit points of a sequence under the continuous map. It may not be compact as the sequence diverges.Any ideas Thanks.
 A: Claim: $A$ is an interval and hence connected.
Proof: Suppose $x,y \in A$ (Wlog assume $x>y)$ by definition of $A$ there exist $(x_n)$ and $(y_n)$ diverging to infinity such that $f(x_n) \to x$ and $f(y_n) \to y$.Suppose $z \in (x,y)$,note that  for $n$ sufficiently large we have $f(x_n)>z>f(y_n)$ (Why?).Hence by Intermediate Mean Value Theorem there exist $z_n$ between  $x_n$ and $y_n$  such that....
Can you complete from here?
A: Addendum: The first two are easily shown to be false by a single example, $f(x)=x\sin(x)$, which leaves only the third.
$A$ could be a singleton as in the case of $f(x)=\arctan(x)$ in which case $A$ would be connected. But suppose $A$ is not a singleton.
Let $p<q$ denote two elements of $A$ and let $\{p_k\}\to \infty$ and $\{q_k\}\to \infty$ such that $\{f(p_k)\}\to p$ and $\{f(q_k)\}\to q$.
Pick an $r$ such that $p<r<q$ and let $0<\epsilon<\min\{r-p,q-r\}$.
Pick a sequence satisfying
$(1)\quad w_0$ is the first element $p_m$ of $\{p_k\}$ satisfying $|p-p_m|<\epsilon$
$(2)\quad w_1$ is the first element $q_m$ of $\{q_k\}$ satisfying $|q-q_m|<\epsilon$ and $w_0<q_m$.
$(3)$ For $n>0$, pick $w_{2n}$ the first element $p_m$ of $\{p_k\}$ satisfying $|p-p_m|<\epsilon$ and $w_{2n-1}<p_m$.
$(4)$  For $n>0$, pick $w_{2n+1}$  the first element $q_m$ of $\{q_k\}$ satisfying $|q-q_m|<\epsilon$ and $w_{2n}<q_m$.
So $\{w_{2n}\}$ is an increasing subsequence of $\{p_n\}$ and $\{w_{2n+1}\}$ is an increasing subsequence of $\{q_n\}$ and $w_{2n}<w_{2n+1}$ for $n\ge0$.
Since $f$ is continuous, for each $n\ge0$ one can pick an $r_n\in(w_{2n},w_{2n+1})$ satisfying $f(r_n)=r$.
Then $\{r_k\}\to\infty$ and $\{f(r_k)\}\to r$.  
So $r\in A$. Therefore $[p,q]\subset A$.
Since $A$ is an arcwise connected subset of $\mathbb{R}$, $A$ is connected.
