Uniform and Pointwise Convergence
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QUESTION 1
For each natural number n and each x in [0,1], define
f_n(x)= x/(nx+1)
Find the function f: [0,1] -> R to which the sequence {f_n} converges pointwise. Prove that the convergence is uniform.
QUESTION 2
For each natural number n and each number x, define
f_n(x)= (1-|x|^n)/(1+|x|^n)
Find the function f: R -> R to which the sequence {f_n} converges pointwise. Prove that the convergence is NOT uniform.
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Solution Summary
Uniform and Pointwise Convergence is examined.
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Question #1
,
We have as . If , then we have as . Now set , then point-wisely. Now I show ...
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