Existence of shortest path in a graph with no negative cycles?
If there is a path $P$ from $s$ to $v$, there is one where no vertex
occurs more than once, with a weight that is less or equal to $P$. Indeed, if a vertex $x$ occurs twice, there is necessarily a loop
from $x$ to itself. This loop cannot have a negative weight by
hypothesis. So by removing it, we get a path that is no heavier. This
can be repeated until we obtain a path where no vertex occurs more
than once. Hence there are at most $n$ vertices on such a path, which
means that the path has at most $n-1$ edges.

Thus we can consider only paths with at most $n-1$ edges, with each
edge occurring at most once, since any other path is no lighter than a
path in that set.

Since this set is finite, the number of edge conbinations to be used for a
path is necessarily finite too (further limited by the constraint or
forming a path from $s$ to $v$). Thus the set of weights has a minimum
corresponding to some path(s) wich is (are) the minimum /shortest path(s).