I learned that a simple path is a path p = $v_0,...,v_m$ with each $v_i \neq v_{i+1}$ with i $\in [m]$ and its a closed path/cycle, if $v_0 = v_m$, so the last node of the path is the same as the beginning.
Now I came across the statement: "In a simple graph exists a simple path between the nodes u and v, if there is a path between u und v."
But that path could also be a loop, if u = v, which is per definition not a simple path.
So why is this statement true?
If $u=v$, then the sequence "$u$" is a simple $u,v$-path with length $m=0$.
(Note that usually, what you are calling a "path" is called a "walk", and what you are calling a "simple path" is just called a "path"; a closed walk with no repeated vertices except the first and last is called a "cycle".)