A line segment is also made up of an infinite set of points, but it's not endless.

I can just imagine someone coming up with the infinity symbol, and arguing (as we are here) about whether a circle represents the idea. No, not quite; it needs something more... another circle should suffice, and connect them seamlessly. Yes, yes. This looks much more infinite than a mere circle.

You can also map the all reals to the reals between 0 and 1, so the infiniteness of the extend isn't that crucial.