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IntroAnalysis
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Homework Statement
Define : R as follows:
For n element J, n >= 2, f(1/n) = 1/(n - 1)
and for all other x element (0, 1), f(x) = x.
Prove that (0,1) is equivalent to [0,1].
Homework Equations
Equivalent means we must prove that (0,1) is 1-1 and onto [0,1].
The Attempt at a Solution
For n=2, we get f(1/2) = 1/(2 -1) = 1 and as n gets larger, 1/(n - 1) approaches 0. Since n
is an integer, 1/n is rational, so let x represent all irrational numbers in (0, 1).
Additionally, suppose f(x1) = f(x2) and x1 does not = x2, then since f(x) = x, we have
x1 = f(x1) = f(x2) = x2, which is a contradiction. Hence f is 1-1 from (0, 1) into [0, 1].