Proof of a lemma used to prove the Schröder – Bernstein theorem [on hold]

Lemma: If g:A→Ag:A \to A is a one-to-one map such that g(A)⊆Cg(A) \subseteq C for some C⊆AC\subseteq A, then there exists a bijection f:A \to Cf:A \to C.

This lemma is used to prove the “Cantor-Bernstein Theorem”. I got a partial proof as follows.

Let X_1 = A \setminus C, \ X_2 = g(X_1), \ X_3 = g(X_2), \dots, X_{n+1} = g(X_n), \dotsX_1 = A \setminus C, \ X_2 = g(X_1), \ X_3 = g(X_2), \dots, X_{n+1} = g(X_n), \dots, for all n \in \Bbb Nn \in \Bbb N, and let X = \bigcup _{i=1} ^\infty X_iX = \bigcup _{i=1} ^\infty X_i.

Now, we define a map f : A \to C, \ f(a)=\begin{cases} g(a), & \text{if } a \in X \\ a, & \text{if } a \in A \setminus C .\end{cases}f : A \to C, \ f(a)=\begin{cases} g(a), & \text{if } a \in X \\ a, & \text{if } a \in A \setminus C .\end{cases}

Claim: ff is injective. I cannot proceed further, help me out of it.

=================

=================

=================