Path-Connectedness of Union

Question

Can I have a hint to prove that $A \cup B$ is not path-connected, where $A = \{(x,y):0 \le x \le 1, y = x/n \text{ for n} \in \mathbb{N} \}$ and $B = \{(x,y):1/2 \le x \le 1, y = 0 \}$?

Answer 1

Assume that $p$ is a path from $(0,0)$ to some point in $B$. As $B$ is closed, we can assume that $p$ doesn't intersect $B$ until the end where $p(1)=:b\in B$. There is a connected neighborhood $U$ around $1\in I$ such that $p[U]\subseteq A\setminus\{(0,0)\}\cup B$. But then $p[U\setminus\{1\}]\subset A\setminus\{(0,0)\}$ which is not connected.

You can think of $A\setminus\{(0,0)\}$ as a topological sum of lines $$A_n=\{(x,y)\mid 0<x\le1, y=x/n \}$$ Since $p[U\setminus\{1\}]$ is connected, it must be contained in some $A_n$. But $p(1)\in B$. Can you reach a contradiction?