Chapter 1.6, Problem 6E

Prove that
(a) for every natural number n, $\frac{1}{n}\le 1$ .
(b) there is a natural number M such that for all natural numbers $n>M$ , $\frac{1}{n}<0.13$ .
(c) for every natural number n, there is a natural number M such that $\text{2}n<M$ .
(d) there is a natural number M such that for every natural number n, $\frac{1}{n}<M$ .
(e) there is no largest natural number.
(f ) there is no smallest positive real number.
(g) For every integer k there exists an integer m such that for all natural numbers n, we have $0\le m+k<n$ .
(h) For every natural number n there is a real number r such that for allnatural numbers m and t, if $t>m>\frac{\text{1}}{r}$ , then $t+n>\text{1}0\text{2}$ .
(i) there is a natural number K such that $\frac{1}{r^2}<0.01$ whenever r is a realnumber larger than K.
(j) there exist integers L and G such that L < G and for every real number x, if $L<x<G$ , then 4 $0>\text{1}0-\text{2}x>\text{12}$ .
(k) there exists an odd integer M such that for all real numbers rlarger thanM, we have $\frac{1}{2r}<0.01$ .
(l) for every natural number x, there is an integer k such that $\text{3}.\text{3}x+k<\text{5}0$ .
(m) there exist integers $x<\text{1}00$ and $y<\text{3}0$ such that $x+y<\text{128}$ and for allreal numbers r and s, if $r>x$ and $s>y$ , then $(r-\text{5}0)(s-\text{2}0)>\text{39}0$ .
(n) for every pair of positive real numbers x and y where $x<y$ , there existsa natural number M such that if n is a natural number and $n>M$ , then $\frac{1}{n}<(y-x)$ .