There are many posts in google about this. I am not from core mathematics background. Deleted infinite broom are lines connecting $(0,0)$ to $(1,\frac{1}{n})$ but point $(1,0)$ is not included.
I know this is NOT path-connected but connected.
If I had to prove that it is not path connected, there should exist 2 points between which no path exists. But in proof why is that $(1,0)$ is considered when it is not part of $X$, where $X$ is the deleted infinite broom.
I understood the definition of path connectedness as any 2 points $x$ and $y$ in $X$, there should be path $P$ (which is continuous function from $[0,1]$ to $X$)
Like mathcounterexamples.net says, the space you describe is in fact path connected. The space known generally as the infinite broom (https://en.wikipedia.org/wiki/Infinite_broom) contains the lines you describe in addition to the interval $(\frac{1}{2},1]$ on the $x$-axis, This space is not path connected because it is impossible to join any point on the interval $(\frac{1}{2},1]$ to any point on one of the other lines making up the broom.