$\textbf{Question:}$ Let $A$ be The deleted infinite broom and let $B$ be the segment $[0.5,1]x{0}$. Prove that $A\cup B$ is not path-connected.
$\textbf{My attempt:}$ For this, I tried to show it's impossible find a path such that $(0,0)$ can be joined to $(1,0)$. However, I don't get it, and I don't know if this way is the most straightforward...
I'll grateful with your help.
$\textbf{NOTES:}$ I'm considering $A$ and $B$ with the induced topology by $R^2$.
This is the definition that I have about The deleted infinite broom: For each integer $n > 0$, let $L_n$ be the line segment in $R^2$ joining the origin $(0,0)$ to the point $(1, \frac{1}{n})$. $A$ would be $\cup L_n \cup (0,1)$
Your overall plan is good, it just needs execution.
Hint: Suppose you had a path $\gamma(t)=(\gamma_x(t),\gamma_y(t))$ such that $\gamma(0) = (0,0)$ and $\gamma(1)=(1,0)$.
Now let $t_0 = \sup \{ t \mid \gamma_x(t) \le \frac13\}$. What can you say about $\gamma(t_0)$? And what can you then say about $\gamma(t)$ for $t>t_0$?