isometry problem, pls help

[minibear]

1. The problem statement, all variables and given/known data

Let (K, d) and (K', d') be two compact metric spaces and let f:K-->K' and g:K'--->K be isometries. Show that f(K)=K' and g(K')=K



2. Relevant equations
n/a

3. The attempt at a solution
I know that isometry implies that I can find one-to-one correspondence mapping, but not sure how to show both function and inverse function are subjective. Please help. Thanks!

[Unco]

1. The problem statement, all variables and given/known data

Let (K, d) and (K', d') be two compact metric spaces and let f:K-->K' and g:K'--->K be isometries. Show that f(K)=K' and g(K')=K

3. The attempt at a solution
I know that isometry implies that I can find one-to-one correspondence mapping, but not sure how to show both function and inverse function are subjective. Please help. Thanks! I can't follow your attempt: we are not looking to 'find' a map. Could you please write out your work in full. It also helps to write out the definitions of the key terms: e.g., isometry, compact.

Are you aware of the theorem which says that an isometry h:X\to X on a compact metric space X is surjective?

If so, the result you want to show is just a corollary of this. If not, then this is a good statement to prove.