Math Problems Vault

Problem 1.8

Use an induction argument to show that for each natural number $n$ , the interval ( $n, n+1$ ) fails to contain any natural number.

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Problem 1.9

Use an induction argument to show that if $n > 1$ is a natural number, then $n-1$ also is a natural number. Then use another induction argument to show that if $m$ and $n$ are natural numbers with $n > m$ , then $n-m$ is a natural number.

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Problem 1.10

Show that for any real number $r$ , there is exactly one integer in the interval $[r, r+1)$ .

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Problem 1.11

Show that any non-empty set of integers that is bounded above has a largest member.

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Problem 1.12

Show that the irrational numbers are dense in $\mathbb {R}$ .

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Problem 1.13

Show that each real number is the supremum of a set of rational numbers and also the supremum of a set of irrational numbers.

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Problem 1.14

Show that if $r > 0$ , then, for each natural number $n,(1+r)^{n} \geq 1+n \cdot r$ .

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Problem 1.15

(i)(ii)(iii)

Use induction arguments to prove that for every natural number $n$ ,

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(i)

$\sum_{j = 1}^{n} j^{2} = \frac{n(n+1)(2 n+1)}{6}$ ,

(ii)

$1^{3}+2^{3}+\ldots +n^{3} = (1+2+\ldots +n)^{2}$ ,

(iii)

$1+r+\ldots +r^{n} = \frac{1-r^{n+1}}{1-r}$ if $r \neq 1$ .