0
$\begingroup$

I want to show if the following sequence of functions converges pointwise or uniformly on $R.$ $f_n(x) = 0$ if $x<n $ and $1/x$ if $x\geq n $. Here's what I did, fix $x\in R $. for all $n \geq 1/x$, $f_n(x) \rightarrow1/x$. Furthermore, $f_n(0)=0$. So $f_n$ converges to $f$ s.t. $f(x) =0$ if $x =0$ and $1/x$ if $0<x\leq 1$. So it's pointwise convergent. Is this correct?

For uniform convergent I say take $x$ s.t. $0<x<n$ then $f_n(x)-f(x) = 0-1/x $ which doesn't $\rightarrow 0 $. So it's not uniformly convergent.

$\endgroup$

1 Answer 1

Reset to default
2
$\begingroup$

Observe that: $$0\leq f_n(x)\leq\frac1n$$ for every $x\in\mathbb R$.

This together with $\lim_{n\to\infty}\frac1n=0$ tells us that $f_n$ converges uniformly (hence also pointwise) to the zero function.

$\endgroup$
2
  • $\begingroup$ So is the limit 0? $\endgroup$
    – Jack
    Commented Feb 28, 2018 at 19:39
  • 1
    $\begingroup$ The limit is the function prescribed by $x\mapsto0$. $\endgroup$
    – drhab
    Commented Feb 28, 2018 at 19:40

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .