Real Analysis Problem

**dhhnerd** · November 11th 2008, 07:21 AM

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.

**ThePerfectHacker** · November 12th 2008, 05:12 PM

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.

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