- #1
complexnumber
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Homework Statement
Define the sequence [tex]\displaystyle f_n : [0,\infty) \to
\left[0,\frac{\pi}{2}\right)[/tex] by [tex]f_n(x) := \tan^{-1}(nx), x \geq
0[/tex].
Homework Equations
Prove that [tex]f_n[/tex] converges pointwise, but not uniformly on
[tex][0,\infty)[/tex].
Prove that [tex]f_n[/tex] converges uniformly on [tex][t, \infty)[/tex] for [tex]t >
0[/tex].
The Attempt at a Solution
[tex]\displaystyle \lim_{n \to \infty} f_n(x) = \lim_{n \to \infty}
\tan^{-1} (nx) = 0[/tex] for [tex]x = 0[/tex].
[tex]\displaystyle \lim_{n \to \infty} f_n(x) = \lim_{n \to \infty}
\tan^{-1} (nx) = \frac{\pi}{2}[/tex] for [tex]x \in (0, \infty)[/tex].
Let [tex]\displaystyle f : [0, \infty) \to \left[0, \frac{\pi}{2}
\right)[/tex] be defined by
[tex]
\begin{align*}
f(x) = \left\{
\begin{array}{ll}
0 & \text{ if } x = 0 \\
\dfrac{\pi}{2} & \text{ if } x > 0
\end{array}
\right.
\end{align*}
[/tex]
Therefore [tex]f_n[/tex] converges pointwise to [tex]f[/tex].
Is function [tex]f[/tex] correct? How can I prove the rest?