I just wanna know if my argument could be written down on my final exam paper as a correct formal argument.

This is the problem: If m<n, show that there is a 1-1 mapping

such that

for all

.

well, my idea is that with every element f in

I associate an element f' in

defined in this way:

If then , otherwise: .
then I define

as

. It's easy to show that this operation is well defined. Now, I define the map

by

.

To see that this map is one-to-one, suppose

. from the definition of

, It's easy to see that

.

Now, with the definition of the function F, It's clear that

and we're done.