# Solution 2a n 2b the empty set 2c the set of subsets

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Solution. 2a: N . 2b: the empty set. 2c: the set of subsets of N , or the set R . 2d: it is not possible: uncountable means not countable, countable means finite or denumerable; it follows that uncountable means infinite and not denumerable.

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4 3. Assume the triangle inequality | a + b | ≤ | a | + | b | , a, b R and prove that || a | - | b || ≤ | a - b | a, b R . Solution. a = ( a - b )+ b = ⇒ | a | = | ( a - b )+ b | ≤ | a - b | + | b | , or | a | ≤ | a - b | + | b | . Subtract | b | : | a | - | b | ≤ | a - b | ( * ) . Repeat for b : b = ( b - a )+ a = ⇒ | b | = | ( b - a )+ a | ≤ | b - a | + | a | , or | b | ≤ | b - a | + | a | , which re-arranged gives -| a - b | ≤ | a | - | b | ( ** ) By combining ( * ) with ( ** ) , we get the result.
5 4. For this number 4 , just provide the examples (no justification is needed). 4a. Give one example of a subset A R which has a supremum but not an infimum. 4b. Give one example of a subset B R which has an infimum, but not a minimum. 4c. Give an example of a set C R which is denumerable, not bounded below with a supremum that is not a maximum. 4d. Give an example of a set D R \ Q (the irrational numbers) for which sup D = 0 is not a maximum. Solution. 4a: A = ( -∞ , 0] . 4b: B = (0 , 1] . 4c: C = the strictly negative rational numbers. 4d: D = the negative irrational numbers.

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6 5. Prove that a subset S R is unbounded if and only if it is not contained in any closed bounded interval.
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