“real: chapter_02” — 2011/5/22 — 23:05 — page6—#6
2-6 Real Analysis
f (t) = g(t). If one of the points x or t lie in P and the other does not,
then the same is true of g(x) and g(t) (here use the observed fact that
x ∈ P if and only if f (x) ∈ P) and so once again, we have g(x) = g(t).
Therefore g is one-to-one. Finally, to see that g maps A onto B, let
y ∈ B. In the event that y ∈ P, we have y = g(y). In the event that y is
not a point of P (y ∈ B\P ⊂ C), y ∈ C and we can write y = f (x) for
some point x which also does not lie in P (x ∈ P ⇒ f (x) ∈ P).
In this
case, we have y = f (x) = g(x) and we conclude that g maps A onto B.
This completes the proof of the lemma.
We now complete the proof of Theorem 2.4.8. Choose a one-to-
one function f from A