hw2 - MATH 444 ELEMENTARY REAL ANALYSIS HOMEWORK 2 Due on...

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MATH 444: ELEMENTARY REAL ANALYSIS HOMEWORK 2 Due on Wednesday, Sep 10 Exercises from the textbook. 1.3: 12 2.2: 16; 17 Out-of-textbook exercises (these are as mandatory as the ones from the textbook) . 1. (a) Let n N and prove that every subset A of { 1 , 2 , ..., n } is finite. Hint : Prove this by induction on n . For the step of induction, let A be a subset of { 1 , 2 , ..., n, n + 1 } and note that B := A \ { n + 1 } is a subset of { 1 , 2 , ..., n } . (b) Conclude from part (a) that any subset of a finite set is finite. 2. Prove that if a set A is denumerable then there is a proper subset B A such that B A , i.e. there is a bijection between
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