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Boundedness is not preserved by pointwise convergence.

Each f_n is bounded (by n) on [0,1], the sequence f_n converges pointwise to f on ]0,1], but f is not bounded on ]0,1]. This counter-example shows that pointwise convergence does not, in general, preserve boundedness.

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GIF animation
Each f_n is bounded (by n) on [0,1], the sequence f_n converges pointwise to f on ]0,1], but f is not bounded on ]0,1].

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