Question
Asked Nov 1, 2019
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2. Assume that (X, dX) and (Y, dY ) are complete spaces, and give X × Y the
metric d defined by
d((x1, y1),(x2, y2)) = dX(x1, x2) + dY (y1, y2)
Show that (X × Y, d) is complete.

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Expert Answer

Step 1

To prove that every Cauchy sequence in the product space XxY converges (under the metric defined in the problem)

Step 2

Statement to be proved

To avoid suffixes
denote the metrics as
(X.p). (Y.4). (Xx Y, d
Let {(x, ybe a Cauchy sequence
for the metric
d(a,b).(x, y) p(x, a) + q(b, y).
To prove:(xy)} converges
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To avoid suffixes denote the metrics as (X.p). (Y.4). (Xx Y, d Let {(x, ybe a Cauchy sequence for the metric d(a,b).(x, y) p(x, a) + q(b, y). To prove:(xy)} converges

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Step 3

First prove that the projections of any Cauchy sequenc...

Pr oof Let &> 0, so3M, such that
d{(xy)(xy,,)}<e,Vm,n2M.
p(x +q(ym=V,) < e,Vm.n 2 M
(by definition (
p(x) and q(y y<, Vm, nz M
(as they are positive quantities)
(2)
help_outline

Image Transcriptionclose

Pr oof Let &> 0, so3M, such that d{(xy)(xy,,)}<e,Vm,n2M. p(x +q(ym=V,) < e,Vm.n 2 M (by definition ( p(x) and q(y y<, Vm, nz M (as they are positive quantities) (2)

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Advanced Math