Metric spaces, making proof formal

[cap.r]

Homework Statement

Suppose that X is a metric Space and supposed that the mapping f:R-->X is continuous. Let C be a closed subset of X and let f(x) belong to C if x is rational. Prove that f(R) is in C.

Homework Equations

Def. a set X is closed if every convergent sequence in X converges within X.
Def. a mapping f:X-->Y is continuous at a point p in X provided that whenever a sequence {p_k} in X converges to p, the image sequence {f(p_k)} converges to f(p)

The Attempt at a Solution

let {u_k} be in Q. and u_k-->u in Q. then the continuity of f implies f(u_k)-->f(u), f(u_k),f(u) are in C since u_k,u are in Q.

Now since f is a cont mapping and f(u_k) must converge to f(u) continuously, f can't just jump over all the irrationals between f(u_k) and f(u). so the denseness of irrationals forces f(R) to be in C.

this last part is in no way formal but makes sense to me... can someone please help me formalize this proof. and fix my logic if i am going wrong anywhere.

 

[Billy Bob]

To help out your logic, why don't you start with "Let r be an element of R. I must show that f(r) is in C." Now do you see what sort of sequence to choose.