91Ó°ÊÓ

Show that a totally bounded set is bounded. Show that the closure of a totally bounded set is totally bounded. Show that every subset of a totally bounded set is again totally bounded.

Show that the union of finitely many compact sets is compact.

Are the following possible? (i) A continuous mapping of \([0,1]\) onto \(\mathbb{R}\). (ii) A uniformly continuous mapping of \((0,1)\) onto \(\mathbb{R}\).

Show that the Taylor series expansion about 0 of the function \(f(s)=e^{s}\) is uniformly convergent in \(\mathbb{R}\).