> 
> If $C\subset \mathbb{R}^p$ is connected and $x$ is a cluster point of
>  $C$, then $C\cup\{x\}$ is connected.
> 
> 
> 

My attempt:

I have been told the easiest way to prove disconnected/connected is by contradiction, so I am going to attempt that here.

Let $A,B$ be a disconnection for $D$, where $D =C\cup\{x\}$. Then we have that $A\cap D$ and $B\cap D$ are non-empty and have union $D$. Now suppose that $x\in A$, then $x\notin B\cap D$ since $A\cap D$ and $B\cap D$ are disjoint. And I get stuck and can't continue any further.