Is the space $X$ is complete?(TRue/False)

Question

Let $f:X\to Y$ be a continuous map between metric spaces. Then $f(X)$ is a complete subset of $Y$ if

1. the space $X$ is compact

2. the space $Y$ is compact

3. the space $X$ is complete

4. the space $Y$ is complete

My trial : i was taking $X =Y =R$,,and i know that R is complete as complete map to complete so the correct option will option 3 that is the space X is complete .

Is my answer is correct or not ? pliz verified and tell me the solution i would be more thankful...

Answer 1

If $X$ is compact, then $f(X)$ is compact too and therefore complete.

All other options are false. Can you see why?