How to Prove Isometries are Surjective on Compact Metric Spaces

[minibear]

Homework Statement

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

Homework Equations

n/a

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]

Homework Statement

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

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.