Why is this map order-preserving? Can somebody explain to me why $f: \mathbb N \rightarrow \{1,1+1,1+1+1,...\}$ where 1 is an identity element of ordered field, preserves order? Intuitivelly I understand that it does, but  I don't know how to justify it?

One of the properties that makes a field ordered is that if $a\le b$ then $a+c\le b+c$.  Take $c=1$ and you have your desired property.  (You also need $0\le 1$ in the ordered field).
