91Ó°ÊÓ

If \(\lim _{z \rightarrow \infty} f(z)=a\), and \(f(z)\) is defined for every positive integer \(n\), prove that \(\lim _{n \rightarrow \infty} f(n)=a\). Give an example to show that the converse is false.

Show that the intersection of an arbitrary collection of closed sets is closed and the union of a finite number of closed sets is closed.

Consider two antipodal points \((x, y, u)\) and \((-x,-y,-u)\) on the Riemann sphere. Show that their stereographic projections \(z\) and \(z^{\prime}\) are related by \(z z^{\prime}=-1\). Give a geometric interpretation.

Let \(s_{n}=\sum_{k=1}^{n} 1 / k !\). Use the Cauchy criterion to show that \(\left\\{s_{n}\right\\}\) converges.

Show that the limit points of a set form a closed set.

Show that the image of the circle \(|z|=\sqrt{3}\) under the stereographic projection is the set of all points \(\left(x_{1}, y_{1}, u_{1}\right)\) in the sphere described by \(x_{1}^{2}+y_{1}^{2}=3 / 4\) and \(u_{1}=1 / 2\)

Show that a monotonic real-valued function of a real variable cannot have uncountably many discontinuities.

Show that \(\bar{A}\), the closure of \(A\), is the smallest closed set containing \(A\).

Show that \(f: A \rightarrow B\) is continuous if and only if for every open set \(O\) relative to \(B, f^{-1}(O)\) is an open set relative to \(A\).

Show that a set is connected if any two of its points can be joined by a polygonal line.