Assigment 1 is due Monday 27th July - hand in answers to the questions 3,4,7 and 9 below. For the Worskshop/Tutorial on 17th July, attempt questions 1,2,5,6 and 8.

The University of Waikato Department of Mathematics

Metric Spaces Exercise and Assignment sheet 1:

math501-09B

Assigment 1 is due Monday 27th July - hand in answers to the questions 3,4,7 and 9 below. For the Worskshop/Tutorial on 17th July, attempt questions 1,2,5,6 and 8.

Metric Spaces:

1. Show that if is a metric on X and > 0 then is also a metric on X.

2. Show that (x; y) :=p

jx yj denes a metric on R.

3. Show that in any metric space (X; ), j(x; y) (y; z)j (x; z) for all x; y; z.

4. Show that if (X; ) is a metric space, so is (X; d) where d(x; y) := (x; y)

1 + (x; y): Hint: you will need to use the property

a + b

1 + a + b a

1 + a + b 1 + b which holds for all a 0; b 0.

5. Derive the parallelogram law in a real Hilbert space i.e for all vectors x; y, jjx + yjj²+ jjx yjj² = 2jjxjj²+ 2jjyjj².

Sequences:

6. If S = f1=n : n 2 Ng show that glb S = 0 and lub S = 1.

7. Let S T be non-empty bounded subsets of R. Prove that lubS lubT and that glbS glbT.

8. Prove that an = ( 1)ⁿ does not converge by (i) nding convergent subsequences converging to dierent limits, and then (ii) showing that it is not a Cauchy sequence.

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9. For each n 2 N let

s_n= 1 + 1

2 + + 1 n:

Prove that s2n sn> ¹₂ and hence show that the sequence does not converge in R using the Cauchy criterion.

Taylor expansions:

10. Compute the Taylor polynomials of degree 1,2,3 and 4 for f(x) = x³+ x 1 about x = 2.

11. Compute the Taylor series for x=(1 + x²) about x = 0 and, using the ratio test or otherwise, nd its region of convergence in R

17th July 2009

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