A connected subset of R is either a single point of an interval. Which is definitely path connected. So it remains to prove that path connectdness implies connectdness, which is a theorem true in any metric space.
the problem is to prove that a subset of real numbers is connected if and only if it is path-connected. the way path-connected is defined in this problem is: a set E in (S,d) where (S,d) is metric space with metric d, is said to be path-connected if every pair of points in E is path-connectable. a pair of points p and q in S are said to be path-connectable if there exists a path f: [a,b] -> S with f(a)=p and f(b)=q, where a continuous function f:[a,b] -> S is called a path (a<b).
i was trying to prove this using the definitions, but found it hard to find much in common between the two definitions. help? thanks in advance!