Give an example of a continuous function on a subset A of and a Cauchy sequence in A in which is not a cauchy sequence in
Follow Math Help Forum on Facebook and Google+
Take 0,1] \longrightarrow \mathbb{R}" alt="f0,1] \longrightarrow \mathbb{R}" /> given by and take . Then is certainly not Cauchy.
Crucial point- this sequence converges to a point that is NOT in A. If the converged to a point, a, in A, since f is continuous on A, would have to converge to f(a), and so be Cauchy.
Last edited by HallsofIvy; December 30th 2009 at 08:34 AM.
View Tag Cloud