Is a connected set union limit point a connected set?

Question

Good night, I'm trying this exercise:

Let $C\subset\mathbb{R}^n$ connected and let $x$ be an accumulation point of $C$. Prove that $C\cup\{x\}$ is connected.

I understand the exercise, geometrically at least, it makes a lot of sense but no luck trying to prove it, by previous examples, I get that, the idea of working with contentedness is by contradiction using the disjoint sets that their union are the disconnected set, but I can't get to use the definition of accumulation point in my proof, any hints or ideas would be appreciated, thanks.

Answer 1

Suppose that $C\cup\{x\}$ is not connected and take $A\cup B$ a separation of $C\cup\{x\}$. As $C$ is connected then $C$ is contained in some part of the separation, say $A$. As $B$ is open and $x$ an accumulation point, what can you conclude?

Answer 2

Suppose $A=C\cup\{x\}$ is disconnected, then there exist open subsets $U$ and $V$ of $\mathbb{R}^n$ such that:

1. $A \subset U \cup V$
2. $A\cap U\neq\emptyset$ and $A\cap V\neq\emptyset$
3. $A\cap U\cap V=\emptyset$

From these three facts above prove that $U\cup V$ is a disconnection of $C$, which contradicts the connectedness of $C$.