Prove that (Y, dY) is not complete by find a Cauchy sequence in (Y, dY) so that it is divergent in (Y, dY)

1. Quizz 6

(1) Prove that the subset K = {(x, y) ∈ R² : x²− y² ≤ 1} of R² is not sequentially compact by constructing a sequence in K which does not have any convergent sub- sequence in K.

(2) Let Y = {(x, y) ∈ R² : x²+ y² < 1} and (Y, d_Y) be the metric subspace of (R², d) defined by Y. Here d is the Euclidean metric on R². Prove that (Y, d_Y) is not complete by find a Cauchy sequence in (Y, d_Y) so that it is divergent in (Y, d_Y).

(3) Let Y be a closed subset of R^p and (Y, dY) be the metric subspace of (R^p, d) defined by Y. Prove that (Y, d_Y) is a complete metric space.

(4) The n-th partial sum of an infinite seriesP∞

n=1a_n of real numbers is defined to be sn=

n

X

i=1

ai = a1+ a2+ · · · + an, for n ≥ 1.

The infinite series

∞

X

n=1

a_n is said to be convergent (in R) if the sequence of partial sums (s_n) is convergent in R. Suppose that there exists a sequence of positive real numbers (bn) such that

(a) |a_n| ≤ b_nfor all n ≥ 1, and (b)

∞

X

n=1

bn is convergent.

Prove that

∞

X

n=1

an is absolutely convergent, i.e.

∞

X

n=1

|a_n| is convergent. (Hint: use the triangle inequality and the Cauchy’s criterion)

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