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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 page 10 #10
2-10 Real Analysis
We now claim that this f is one-to-one.
Let f (x) = f (y)(x, y S B).
Case 1: x, y P or x, y S\P or x, y B.
In this case, it is clear that x = y.
Case 2: x S\P, y P or x P, y S\P.
This cannot happen because in this case x = f (x) = f (y) = x
2i
for
some i and x S\P and x
i
P or vice versa.
Case 3: x P, y B,orx B, y P.
We merely assume that x P, y B. The other case can be handled
similarly. In this case, x
2n
= f (x
n
) = f (x) = f (y) = f (y
m
) = x
2m1
for some n and m. In this case also x
2n
= x
2m1
, a contradiction.
Case 4: x S\P, y B (or x B, y S\P). In this case also, we
have x = f (x) = f (y) = f (y
n
) = x
2n1
P, a contradiction to the
fact that
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Publisher Resources

ISBN: 9781299447561Publisher Website