2.2.E: Problems on Natural Numbers and Induction (Exercises)
Complete the missing details in Examples \((\mathrm{a}),(\mathrm{b}),\) and \((\mathrm{d})\).
Prove Theorem 2 in detail.
Suppose \(x_{k}<y_{k}, k=1,2, \ldots,\) in an ordered field. Prove by induction on \(n\) that
(a) \(\sum_{k=1}^{n} x_{k}<\sum_{k=1}^{n} y_{k}\)
(b) if all \(x_{k}, y_{k}\) are greater than zero, then
\[
\prod_{k=1}^{n} x_{k}<\prod_{k=1}^{n} y_{k}
\]
Prove by induction that
(i) \(1^{n}=1\);
(ii) \(a<b \Rightarrow a^{n}<b^{n}\) if \(a>0\).
Hence deduce that
(iii) \(0 \leq a^{n}<1\) if \(0 \leq a<1\);
(iv) \(a^{n}<b^{n} \Rightarrow a<b\) if \(b>0 ;\) proof by contradiction.
Prove the Bernoulli inequalities: For any element \(\varepsilon\) of an ordered field,
(i) \((1+\varepsilon)^{n} \geq 1+n \varepsilon\) if \(\varepsilon>-1\);
(ii) \((1-\varepsilon)^{n} \geq 1-n \varepsilon\) if \(\varepsilon<1 ; n=1,2,3, \ldots\)
For any field elements \(a, b\) and natural numbers \(m, n,\) prove that
\[
\begin{array}{ll}{\text { (i) } a^{m} a^{n}=a^{m+n} ;} & {\text { (ii) }\left(a^{m}\right)^{n}=a^{m n}} \\ {\text { (iii) }(a b)^{n}=a^{n} b^{n} ;} & {\text { (iv) }(m+n) a=m a+n a} \\ {\text { (v) } n(m a)=(n m) \cdot a ;} & {\text { (vi) } n(a+b)=n a+n b}\end{array}
\]
[Hint: For problems involving two natural numbers, fix \(m\) and use induction on \(n ]\).
Prove that in any field,
\[
a^{n+1}-b^{n+1}=(a-b) \sum_{k=0}^{n} a^{k} b^{n-k}, \quad n=1,2,3, \ldots
\]
Hence for \(r \neq 1\)
\[
\sum_{k=0}^{n} a r^{k}=a \frac{1-r^{n+1}}{1-r}
\]
(sum of \(n\) terms of a geometric series).
For \(n>0\) define
\[
\left(\begin{array}{l}{n} \\ {k}\end{array}\right)=\left\{\begin{array}{ll}{\frac{n !}{k !(n-k) !},} & {0 \leq k \leq n} \\ {0,} & {\text { otherwise }}\end{array}\right.
\]
Verify Pascal's law,
\[
\left(\begin{array}{l}{n+1} \\ {k+1}\end{array}\right)=\left(\begin{array}{l}{n} \\ {k}\end{array}\right)+\left(\begin{array}{c}{n} \\ {k+1}\end{array}\right).
\]
Then prove by induction on \(n\) that
(i) \((\forall k | 0 \leq k \leq n)\left(\begin{array}{l}{n} \\ {k}\end{array}\right) \in N ;\) and
(ii) for any field elements \(a\) and \(b\),
\[
(a+b)^{n}=\sum_{k=0}^{n}\left(\begin{array}{l}{n} \\ {k}\end{array}\right) a^{k} b^{n-k}, \quad n \in N \text { (the binomial theorem). }
\]
What value must \(0^{0}\) take for (ii) to hold for all \(a\) and \(b ?\)
Show by induction that in an ordered field \(F\) any finite sequence \(x_{1}, \ldots, x_{n}\) has a largest and a least term (which need not be \(x_{1}\) or \(x_{n} ) .\) Deduce that all of \(N\) is an infinite set, in any ordered field.
Prove in \(E^{1}\) that
(i) \(\sum_{k=1}^{n} k=\frac{1}{2} n(n+1)\);
(ii) \(\sum_{k=1}^{n} k^{2}=\frac{1}{6} n(n+1)(2 n+1)\);
(iii) \(\sum_{k=1}^{n} k^{3}=\frac{1}{4} n^{2}(n+1)^{2}\);
(iv) \(\sum_{k=1}^{n} k^{4}=\frac{1}{30} n(n+1)(2 n+1)\left(3 n^{2}+3 n-1\right)\).