By Hugo D. Junghenn

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Assume that an → a ∈ R. Choose N such that |an − a| < 1 for all n > N . Since |an | − |a| ≤ |an − a|, we see that |an | ≤ |an − a| + |a| < 1 + |a| for all n > N . Thus |an | ≤ max{1 + |a|, |a1 |, . . , |aN |} for all n ∈ N. 4 Theorem. Let {an } and {bn } be sequences with an → a and bn → b. If an ≤ bn for infinitely many n, then a ≤ b. Proof. Suppose b < a. Then b < (a + b)/2 < a, hence we may choose indices N1 and N2 such that bn < (a + b)/2 for all n ≥ N1 and an > (a + b)/2 for all n ≥ N2 . But then bn < an for all n ≥ max{N1 , N2 }, contradicting the hypothesis.

Show that Q( 2) is not complete. S (a) Find all n ∈ N such that n + 11 + n ∈ Q. √ √ (b) Same question for n + 21 + n. 21. Let √ p ∈ N√be prime, that is, divisible only by 1 and itself. Prove that ( n + 1)( n + p + 1)−1 ∈ Q iff n = (p − 1)2 /4. 2. 5 19 Mathematical Induction In this section we give an abstract characterization of the natural number system. This will lead directly to the principle of mathematical induction. 1 Definition. A set S of real numbers is said to be inductive if • 1 ∈ S, • x ∈ S implies x + 1 ∈ S.

If a/x ≤ x + 1 for every x > 0, then a ≤ 0. 13. For all x, y, z, w ∈ R, (a) 2xy ≤ x2 + y 2 . (b) S xy + yz + xz ≤ x2 + y 2 + z 2 . (c) (xy + zw)2 ≤ (x2 + z 2 )(y 2 + w2 ). (d) (x + y)2 ≤ 2(x2 + y 2 ). S If x, a > 0, then x + a2 /x ≥ 2a. Equality holds iff x = a. 15. (a) |x − y| ≤ |x − z| + |z − y|. (b) |x − L| < ε iff L − ε < x < L + ε. 16. Let S, T ⊆ R be finite and nonempty. Define −S := {−s : s ∈ S}. Then (a) max(−S) = − min S. (b) min(−S) = − max S. (c) max(S ∪ T ) = max{max S, max T }. (d) min(S ∪ T ) = min{min S, min T }.

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