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Answered Same Day Dec 22, 2021

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Robert answered on Dec 22 2021

119 Votes

1
1. This is a standard example in topology. You can find the solution here at this
http:
math.stanford.edu/~conrad/diffgeomPage/handouts/sinecurve.pdf
2. No. The given series doesn’t converge uniformly. It converges pointwise to
∞∑
n=1
fn(x1, x2) =
∞∑
n=1
1
n2
· x1
1 + x1 + x2
=
x1
1 + x1 + x2
·
∞∑
n=1
1
n2
=
x1
1...

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