I am looking to find a sequence of functions $f_n$ that converges to a function $f$ pointwise, where all functions $f_n$ are bounded, but $f$ is unbounded.

I have thought of an example where the sequence of function $f_n$: (0,1) $\rightarrow$ $R$ defined by:

$f_n$$(x)$ = $\frac{n}{nx+1}$. I believe that this converges pointwise to the limit $f(x)$= $\frac{1}{x}$ and that each $f_n$ is bounded on (0,1), but $f(x)$ is unbounded.

Is this correct? Thank you!

  • 2
    $\begingroup$ Yes, it is correct. $\endgroup$ – David Mitra Apr 25 '15 at 16:07
  • $\begingroup$ Great, thank you sir! $\endgroup$ – michaelbaes Apr 25 '15 at 16:09

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