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I've read the definition a few times and I am still struggling to understand what these actually mean. As I understand it, a sequence of functions is pointwise bounded if there exists an $M$ for each $x$ such that $f_n(x)<M$, for all $n$, and a function is uniformly bounded if there exists an $M$ such that $f_n(x) < M$ for every $n, x$. This leads to two questions that I have. Firstly, would pointwise convergence imply pointwise boundedness and would uniform convergence imply uniform boundedness? Also are there a few examples that demonstrate some pointwise/uniformly bounded sequences of functions(non trivial, i.e not a constant sequence), particularly perhaps one that is pointwise but not uniformly bounded and why that is the case

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Yes it is true that pointwise convergence implies pointwise bounded. A proof is similar to a proof that the convergent sequence of numbers $f_i(x)$ in the index $i$ is bounded for a fixed $x$, except you apply the $\forall x$ quantifier.

No uniform convergence does not imply uniform boundedness. Take $f_i(x)=x^2+1/i$, they converge to $f(x)=x^2$ uniformly, which is not a uniformly bounded function.

The sequence $f_i$ above is pointwise bounded but not uniformly bounded. Given $x$, a bound for the sequence of numbers $f_i(x)$ is $x^2+1$. It is not uniformly bounded because not even any single $f_i$ is bounded.

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  • $\begingroup$ Your sequence converges to $x^2$ $\endgroup$
    – GTOgod
    Dec 17 '17 at 9:49
  • $\begingroup$ yes oops I saw it and fixed it, thanks for the catch $\endgroup$ Dec 17 '17 at 9:49

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