MATH0051: Analysis 4 — Real Analysis 2022
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MATH0051: Analysis 4 — Real Analysis
Resit coursework 2022
Answer all questions
1. Let X = (0, ∞) and
d(x,y) = I log ( 
) I.
(i) Prove that (X,d) is a metric space;
(ii) Describe all convergent sequences in (X,d).
(iii) Describe all Cauchy sequences in (X,d).
(iv) Is this metric space complete?
Justify your answers.
2. Prove that
||x||∞ = 
|xi |
defines a norm on Rn .
3. Let {an }
be a sequence of positive real numbers such that an → 0 as n → ∞ . Let fn (x) = (x + an )2 , x ∈ R. Does the sequence {fn }
converge pointwise on R? If it does, what is the limit function? Does the sequence {fn }
converge uniformly on
R? Justify your answers. [10 marks]
2022-08-27