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ADVANCED MATH
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ADVANCED MATH
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ADVANCED MATH
Prove that (a) for every natural number n, $\dfrac{1}{n}\leq 1$. (b) there is a natural number M such that for all natural numbers n $>$ M, $\dfrac{1}{n}<0.13$. (c) for every natural number n, there is a natural number M such that 2n $<$ M. (d) for every natural number n, $\dfrac{1}{n}\< M$. (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 $0\leq m+5<n$. (h) For every natural number n there is a real number r such that for all natural numbers m and t, if $t>m>\dfrac{1}{r}$ then $t+n>102$. (i) there is a natural number K such that $\dfrac{1}{r^2} < 0.01$ whenever r is a real number larger than K. (j) there exist integers L and G such that L $<$G and for every real numberx, if $L < x < G$, then $40 > 10 - 2x > 12$. (k) there exists an odd integer M such that for all real numbers r larger than 1 M, we have $\dfrac{1}{2r}< 0.01$. (l) for every natural number x, there is an integer k such that 3.3x + k $<$50. (m) there exist integers $x < 100$ and $y < 30$ such that $x + y < 128$ and for all real numbers rands, if $r > x$ and $s > y$, then $(r - 50)(s - 20) > 390$. (n) for every pair of positive real numbers x and y where $x < y$, there exists a natural number M such that if n is a natural number and $n > M$, then $\dfrac{1}{n}<$ (y - x).
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ADVANCED MATH
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