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Therefore, uniform convergence implies pointwise convergence. But the con-verse is false as we can see from the following counter-example. Example 10 Let {fn} be the sequence of functions on (0, ∞) defined by fn(x) = nx 1+n2x2. This sequence converges pointwise to zero. Indeed, (1 + n2x2) ∼ n2x2 as n gets larger and larger. So, lim n→∞ ...

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