uniformly convergent

**Camille91** · August 31st 2011, 03:59 AM

Let $f_n : \mathbb{R} \rightarrow \mathbb{R}$ be a sequence of functions that is unifromly convergent in $\mathbb{R}$ to funcion $f: \mathbb{R} \rightarrow \mathbb{R}$ . For $n \in \mathbb{N}$ we define:
$g_n(x)=\exp (-(f_n(x))^2), \ \ \ g(x)=\exp(-(f(x))^2)$
$h_n(x)=(f_n(x))^2, \ \ \ h(x)=(f(x))^2$ .

Does sequence $g_n$ uniformly convergent in $\mathbb{R}$ to function $g$ ?
Does sequence $h_n$ uniformly convergent in $\mathbb{R}$ to function $h$ ?

**girdav** · August 31st 2011, 04:58 AM

For the first question, write $e^{-f_n(x)^2}-e^{-f(x)^2}$ as $\int_{f(x)}^{f_n(x)}-2te^{-t^2}dt$ and use the fact that the map $t\mapsto te^{-t^2}$ is bounded on the real line.
For the second question, consider $f_n(x)=x+\frac 1n$ .

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