i am stuck in proving this Theorm according to my textbook .
Theorm States that if $\lambda$ is a ordinal numbers and $S(\lambda)$ be the set of ordinals less than ordinal $\lambda$ . Then ordinal of $S(\lambda)$ is equal to $\lambda$
So proof following a very standered way is through finding a similar mapping between $A$ and $S(A)$ where A is well ordered set whose ordinal number is $\lambda$. and the process of proving it is very easy and i understand it.
But i also studied in basic properties of of well ordered set that A set never similar to it's any of the initial sagment and i also prove this result by contradiction.
So how these two well proved Theorms are opposing to each other. Can anyone please help me to finding flaw in my procedure.