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372 Appendix 1: Solutions to Selected Exercises
f (x) = lim n→∞ f (x n ) = lim n→∞ g(x n ) = g(x). Since x was arbitrary,
Q
this shows that f = g. Therefore F is one-to-one, so C R. Com-
bining this with part (b), we can conclude that C R.
Now define G : R → C by the formula G(x) = R ×{x}. In other
words, G(x) is the constant function whose value at every real number
is x. Clearly G is one-to-one, so R C. By the Cantor–Schr¨oder–
Bernstein Theorem, it follows that C ∼ R.
