Chapter 2

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).

Let \(A\) and \(B\) be subsets of an ordered field \(F\). Assuming that the required lub and glb exist in \(F\), prove that (i) if \((\forall x \in A)(\forall y \in B) x \leq y,\) then \(\sup A \leq \inf B ;\) (ii) if \((\forall x \in A)(\exists y \in B) x \leq y,\) then sup \(A \leq \sup B\) (iii) if \((\forall y \in B)(\exists x \in A) x \leq y,\) then \(\inf A \leq \inf B\).

Prove that if \(\left\\{x_{n}\right\\}\) and \(\left\\{y_{n}\right\\}\) are bounded in \(E^{1}\), then \(\overline{\lim } x_{n}+\overline{\lim } y_{n} \geq \overline{\lim }\left(x_{n}+y_{n}\right) \geq \overline{\lim } x_{n}+\underline{\lim } y_{n}\) \(\geq \underline{\lim }\left(x_{n}+y_{n}\right) \geq \underline{\lim } x_{n}+\underline{\lim } y_{n}\).

For any two subsets \(A\) and \(B\) of an ordered field \(F,\) let \(A+B\) denote the set of all sums \(x+y\) with \(x \in A\) and \(y \in B ;\) i.e., $$ A+B=\\{x+y \mid x \in A, y \in B\\} $$ Prove that if \(\sup A=p\) and \(\sup B=q\) exist in \(F,\) then $$ p+q=\sup (A+B) $$ similarly for infima. [Hint for sup: By Theorem 2 , we must show that (i) \((\forall x \in A)(\forall y \in B) x+y \leq p+q\) (which is easy) and \(\left(\mathrm{ii}^{\prime}\right)(\forall \varepsilon>0)(\exists x \in A)(\exists y \in B) x+y>(p+q)-\varepsilon\) Fix any \(\varepsilon>0 .\) By Theorem 2 , $$ (\exists x \in A)(\exists y \in B) \quad p-\frac{\varepsilon}{2}\left(p-\frac{\varepsilon}{2}\right)+\left(q-\frac{\varepsilon}{2}\right)=(p+q)-\varepsilon $$ as required.]

Prove that if \(p=\lim x_{n}\) in \(E^{1}\), then $$\underline{\lim }\left(x_{n}+y_{n}\right)=p+\underline{\lim } y_{n} ;$$ similarly for \(\bar{L}\).

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) \((\forall k \mid 0 \leq k \leq n)\left(\begin{array}{l}n \\\ k\end{array}\right) \in N ;\) and (ii) for any field elements \(a\) and \(b\), \((a+b)^{n}=\sum_{k=0}^{n}\left(\begin{array}{l}n \\ k\end{array}\right) a^{k} b^{n-k}, \quad n \in N\) (the binomial theorem). What value must \(0^{0}\) take for (ii) to hold for all \(a\) and \(b ?\)

Prove that (i) if \((\forall \varepsilon>0) a \geq b-\varepsilon,\) then \(a \geq b ;\) (ii) if \((\forall \varepsilon>0) a \leq b+\varepsilon,\) then \(a \leq b\).

Prove that (i) if \(\lim x_{n}=+\infty\) and \((\forall n) x_{n} \leq y_{n},\) then also \(\lim y_{n}=+\infty,\) and (ii) if \(\lim x_{n}=-\infty\) and \((\forall n) y_{n} \leq x_{n},\) then also \(\lim y_{n}=-\infty\).

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) \(\sum_{k=1}^{n} k^{4}=\frac{1}{30} n(n+1)(2 n+1)\left(3 n^{2}+3 n-1\right)\).

Prove the principle of nested intervals: If \(\left[a_{n}, b_{n}\right]\) are closed intervals in a complete ordered field \(F\), with $$ \left[a_{n}, b_{n}\right] \supseteq\left[a_{n+1}, b_{n+1}\right], \quad n=1,2, \ldots $$ then $$ \bigcap_{n=1}^{\infty}\left[a_{n}, b_{n}\right] \neq \emptyset $$ [Hint: Let $$ A=\left\\{a_{1}, a_{2}, \ldots, a_{n}, \ldots\right\\} $$ Show that \(A\) is bounded above by each \(b_{n}\). Let \(p=\sup A\). (Does it exist?) Show that $$ (\forall n) \quad a_{n} \leq p \leq b_{n} $$ i.e., $$ \left.p \in\left[a_{n}, b_{n}\right] \cdot\right] $$