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Prove or disprove that there exists a bijection from (0, 1] to [0, ∞)^2.
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Define f:(0,1] →R as follows.
For n∈N, n≥2, f(1/n)=1/(n−1) and for all other x∈(0,1] , f(x)=x
- Prove that f is a 1−1 function from (0,1] onto [0, ∞)2
- Slightly modify the above function to prove that (0,1] is equivalent to [0, ∞)2
- Prove that (0,1] is equivalent to [0, ∞)2
Since the "equivalent to" relation is both symmetric and transitive, it should follow that (0,1] is equivalent to [0,∞)2. Hence, there does exist a one-to-one correspondence between (0,1] and [0, ∞)2