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If m > m, then there is some x 2 E such that x < m . the value of m, then parts 1 and 2. Notice that, if m is a lower bound of E and m 2 E, then part 2 is automatically satisfied, and so m ¼ inf E ¼ min E. Problem 5 Determine inf E, if it exists, for each of the following sets: (a) E1 ¼ (1, 5]; (b) E2 ¼ fn12 : n ¼ 1; 2; . g. Remarks 1. For any subset E of R, inf E sup E. This follows from the fact that, for any x 2 E, we have inf E x sup E. 2. For any bounded interval I of R, let a be its left end-point and b its right end-point.

STEP 2 We now assume that P(k) holds for some k ! 4, and deduce that P(k þ 1) is then true. So, we are assuming that 2k ! k2. Multiplying this inequality by 2 we get 2kþ1 ! 2k2 ; so it is therefore sufficient for our purposes to prove that 2k2 ! (k þ 1)2. Now 2k2 ! ðk þ 1Þ2 , 2k2 ! k2 þ 2k þ 1 , k2 À 2k À 1 ! 0 ðby ‘completing the square’Þ , ðk À 1Þ2 À2 ! 0: This last inequality certainly holds for k ! 4, and so 2kþ1 ! (k þ 1)2 also holds for k ! 4. In other words: P(k) true for some k ! 4 ) P(k þ 1) true.

Find x þ y and xy (in decimal form). 4. 12). pffiffiffi 5. Prove that, if n is a positive integer which is not a perfect square, then n is irrational. ), p show that 0 < p À kq < q and nqÀkp pÀkq ¼ q, and hence obtain a contradiction. 2 1. Solve the following inequalities: pffiffiffiffiffiffiffiffiffiffiffiffiffi xþ1 (a) xxÀ1 (b) 4x À 3 > x; 2 þ4 < x2 À4;   (c) 17 À 2x4  15; (d) jx þ 1j þ jx À 1j < 4. 3 1. Use the Triangle Inequality to prove that jaj 1 ) ja À 3j ! 2: 2. Prove that (a2 þ b2)(c2 þ d2) ! (ac þ ad)2, for any a, b, c, d 2 R.

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