Problem 3
Suppose \(x_{k}
Problem 3
Find \(\overline{\lim } x_{n}\) and \(\lim x_{n}\) if (a) \(x_{n}=c\) (constant); (b) \(x_{n}=-n ;\) (c) \(x_{n}=n ;\) and (d) \(x_{n}=(-1)^{n} n-n .\) Does \(\lim x_{n}\) exist in each case?
Problem 4
Prove that if \(A=(a, b)\) is an open interval \((a
Problem 4
Prove that if \(r\) is rational and \(q\) is not, then \(r \pm q\) is irrational; so also are \(r q, q / r,\) and \(r / q\) if \(r \neq 0\) [Hint: Assume the opposite and find a contradiction.]
Problem 5
Prove that (i) \(\overline{\lim }\left(-x_{n}\right)=-\underline{\lim } x_{n}\) and (ii) \(\overline{\lim }\left(a x_{n}\right)=a \cdot \overline{\lim } x_{n}\) if \(0 \leq a<+\infty\).
Problem 5
In an ordered field \(F,\) let \(\emptyset \neq A \subset F\). Let \(c \in F\) and let \(c A\) denote the set of all products \(c x(x \in A) ;\) i.e., $$ c A=\\{c x \mid x \in A\\} $$Prove that (i) if \(c \geq 0\), then $$ \sup (c A)=c \cdot \sup A \text { and } \inf (c A)=c \cdot \inf A $$ (ii) if \(c<0\), then $$ \sup (c A)=c \cdot \inf A \text { and } \inf (c A)=c \cdot \sup A $$ In both cases, assume that the right-side sup \(A\) (respectively, inf \(A\) ) exists.
Problem 5
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 \(\varepsilon<1 ; n=1,2,3, \ldots\)
Problem 6
For any field elements \(a, b\) and natural numbers \(m, n,\) prove that (i) \(a^{m} a^{n}=a^{m+n}\) (ii) \(\quad\left(a^{m}\right)^{n}=a^{m n} ;\) (iii) \(\quad(a b)^{n}=a^{n} b^{n} ;\) (iv) \(\quad(m+n) a=m a+n a ;\) (v) \(n(m a)=(n m) \cdot a ;\) (vi) \(\quad n(a+b)=n a+n b\).
Problem 6
Prove that the rational subfield \(R\) of any ordered field is Archimedean. [Hint: If $$ x=\frac{k}{m} \text { and } y=\frac{p}{q} \quad(k, m, p, q \in N), $$ then \(n x>y\) for \(n=m p+1]\)
Problem 6
Prove that \(\overline{\lim } x_{n}<+\infty\left(\underline{\lim } x_{n}>-\infty\right)\) iff \(\left\\{x_{n}\right\\}\) is bounded above (below) in \(E^{1}\).
