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Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“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
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Publisher Resources

ISBN: 9781299447561Publisher Website