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MATH 401 INTRODUCTION TO ANALYSIS,

SPRING TERM 2016, PROBLEMS 7

Return by Monday 29th February

1. Let a be any element of the open interval (2, 3).

(i) Show that there is another b ? (2, 3) with b &gt; a.

(ii) Prove that (2, 3) has no maximum.

2. Suppose that A is bounded above, B ? A and B is non-empty. Prove that sup B

exists and sup B ? sup A.

3. Let A = {x : x ? Q, x3 &lt; 2}. Prove that sup A exists. Guess the value of sup A.

4. Suppose that x ? ?1.

(i) Prove that (1 + x)2 ? 1 + 2x.

(ii) Prove that (1 + x)3 ? 1 + 3x.

Guess a general inequality for (1 + x)n when x ? ?1 and n ? N.

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