given two points p, q in a connected metric space and $\epsilon > 0$
I would like to show there exists pts $p=a_0,....,a_n=q$ s.t. the $d_X(a_i,a_{i-1})< \epsilon, \ \forall i\in{1,...,n}$
My idea is to say the distance function $d_X(p,x)$ is continuous and the continuous image of a connected set is connected. Then split up the range into $n$, $\epsilon$ sized intervals then take one pt from each interval and let a_i be the pre-image of that point.
I know $d_X(p,p)=0$ and $d_X(p,q) \geq 0$ so the distance must take on all the values in between.
So I think my only hangup is how to show the distance function is continuous.