Hi,can anyone please help me with this question? I cannot even start it.
Let f: be a continuous function and let
Prove that C is a closed subset of .
Alternatively you could also prove it with the sequences definition of closed sets in :
is closed iff the limit of every convergent sequence , with , stays in A.
Let be any sequence in C, which has a limit x.
For all n
=> , because f is continous => .
qed
Btw. more generally for any continous function f, it is true that is always a closed set, when A is a closed set and is always an open set, when A is an open set.
In your exercise, we just have the special case , which is closed and so must be closed too.