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Homework Problem 1. Let {fn} be the sequence of functions on [0, 1] defined by fn(x) = nx(1x4)n. Show that {fn} converges pointwise. Find its...

Homework
Problem 1.
Let {fn} be the sequence of functions on [0, 1] defined by fn(x) = nx(1−x4)n.
Show that {fn} converges pointwise. Find its pointwise limit.
Problem 2.
Is the sequence of functions on [0, 1) defined by fn(x) = (1 − x)
1
n pointwise
convergent? Justify your answer.
Problem 3.
Consider the sequence {fn} of functions defined by
fn(x) =
n + cos(nx)
2n + 1
for all x in R.
Show that {fn} is pointwise convergent. Find its pointwise limit.
Problem 4.
Consider the sequence {fn} of functions defined on [0, ] by fn(x) = sinn(x).
Show that {fn} converges pointwise. Find its pointwise limit. Using the
above theorem, show that {fn} is not uniformly convergent.
5

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