I am struggling with the question below:
Suppose that $ C \in \Re^m $ is a compact set and $f: C \rightarrow \Re^n$ is a continuous function. Prove that $f(C) := \{y|y=f(x) \text{ for some } x \in C \}$ is a closed set.
How can I prove that $ f(C) $ is closed? I have tried the following:
- To show it directly using the definition of the closed set.
- To find some contradictions resulting from assuming $f(c)$ is not closed.
... which simply made no avail thanks to my limited knowledge in topology.
I looked up the answers from the "similar questions" tab, but I couldn't understand most of the discussion. I am an Econ undergraduate, and know only the very basics of topology. (i.e. the only metric spaces I am familiar with is the Euclidean metric, etc.)
How can I prove this? Even the slightest help in any form would be appreciated.