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Cantor). Prove that |A| ≤ |P(A)| and |A| = |P(A)| for every set A. In particular, P(N) is uncountable. [Hint: The nontrivial case is where A is nonempty. Then the function f (x) = {x} provides a one-to-one map from A to P(A), hence |A| ≤ |P(A)|. Assuming the existence of a bijection g : A → P(A), we have to consider the set X = {a : a ∈ A and a ∈ / g(a)}. ] According to Exercise 4, |P(N)| = 2N . 8 below and the Cantor-Schröder-Bernstein Theorem, one can infer that 2N = c. Therefore |P(N)| = c. 6.

It is useful to extend the algebraic structure of R, by defining some operations with infinite elements. More precisely, the addition is supplemented by x + (−∞) = (−∞) + x = −∞ for all x ∈ R x + ∞ = ∞ + x = ∞ for all x ∈ R (−∞) + (−∞) = −∞ and ∞ + ∞ = ∞, while multiplication is supplemented by x · (−∞) = (−∞) · x = x ·∞=∞·x = ∞ −∞ −∞ ∞ if x ∈ R, x < 0 if x ∈ R, x > 0 if x ∈ R, x < 0 if x ∈ R, x > 0. All these operations are motivated by their companions in terms of limits. See Exercise 2. 54 2 Limits of Real Sequences We do not define ∞ − ∞, (−∞) + ∞, 0 · (−∞), (−∞) · 0, 0 · ∞, ∞ · 0 nor 0 , 0 ∞ , 1∞ , ∞0 and 00 .

2) for all nonnegative integers k. We call (dn−k )k the sequence of digits of x; dn−k represents the digit of order 10n−k . The digits of negative order are called decimals. The mapping x → (dn−k )k is injective. 6 we conclude that x − y = 0. An important remark is the nonexistence of an integer N such that dn−k = 9 for all k ≥ N. 5 The Decimal Representation z+ z+ 27 9 10 ≤x