Prove that the map f : C → R 2 , x + iy 7→ (x, y) is a bijection, i.e., a cardinal equivalence C ≈ R
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Let us prove that the map is a bijection.
Let Then and hence and We conclude that and thus the map is injective.
Let be arbitrary. Then for we get that and consequently, is surjective.
We conclude that is a bijection.