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We have sequence of function $\{f_n(x)\}$ defined on set $E$ and $n\in \mathbb{N}$. What does mean sequence of bounded functions? Is it pointwise bounded?

Definition: We say that $\{f_n\}$ is pointwise bounded on $E$ if the sequence $\{f_n(x)\}$ is bounded for every $x\in E$, that is, if there exists a finite-valued function $\phi$ defined on $E$ such that $$|f_n(x)|<\phi(x) \quad (x\in E, n\in \mathbb{N}).$$

Can anyone explain to me please?

  • $\begingroup$ just to add.why it is NOT pointwise bounded because once we know that it has uniform convergence it implies pointwise boundedness $\endgroup$ – viru Nov 20 '19 at 12:40

A bounded function is a function whose range is contained in a finite interval. A sequence of bounded functions has the property that each of its individual functions $f_n$ has such a limited range (but the overall range of the collection may still be infinitely large).

The sequence is pointwise bounded if for every $x$ separately the sequence $f_n(x)$ is contained in a finite interval. That is a different condition.

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  • $\begingroup$ So sequence of bounded functions $\{f_n(x)\}$ means that $|f_n(x)|<M_n$ for any $n$. Right? $\endgroup$ – ZFR Nov 26 '15 at 20:06
  • $\begingroup$ Exactly. And pointwise bounded sequence means that $|f_n(x)|<m_x$ for any $x.$ $\endgroup$ – Justpassingby Nov 26 '15 at 20:11

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