# Real Analysis Problem

• Nov 11th 2008, 07:21 AM
dhhnerd
Real Analysis Problem
Let f: (0,1] -> R be differentiable on (0,1], with |f(x)|<=1 for all x in (0,1]. For each n in N, let a(sub n)=f(1/n). Show that a(sub n), with n in N, converges.
• Nov 12th 2008, 05:12 PM
ThePerfectHacker
Quote:

Originally Posted by dhhnerd
Let f: (0,1] -> R be differentiable on (0,1], with |f(x)|<=1 for all x in (0,1]. For each n in N, let a(sub n)=f(1/n). Show that a(sub n), with n in N, converges.

Let $f(x) = \sin \tfrac{1}{x}$ then $f$ is differenciable on $(0,\infty)$ and $|f|\leq 1$. The sequence is $a_n = f(\tfrac{1}{n}) = \sin (n)$.
However, $\{a_n\}$ is not a convergent sequence.