Mathematics 2C - Introduction to Real Analysis 2021
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Mathematics 2C - Introduction to Real Analysis
2021
1. (i) Show that the set
A = {5x − 7y + | x,y,z ∈ ( −3, − 1) }
is bounded.
(ii) Show that
sup { 6(5)n(n) 4(3) | n ∈ N} =
2.
Prove the following statements directly from the definition.
(i) If (xn ) and (yn ) are real sequences with limits L and M , respectively, then xn − 3yn → L − 3M as n → ∞ .
(ii) The function f : R → R,f(x) = 3x2 − 3x + 1 is continuous at x = 1.
3. (i) Prove that the sequence xn = (3n + 3)1/n is convergent.
(ii) Suppose (xn ) is a real sequence such that xn 1 for all n ∈ N and → 2 as n → ∞ . Determine if (xn ) converges, and if so, compute the value of its limit.
4. For each of the series below, determine whether they converge or diverge. Justify your answers clearly, referring to any results or tests you use from the course. Any answer with no justification will receive no marks.
(i)
( ) 1/n .
(ii)
n(n)3−
(iii)
( − 1)n .
5. (i) Is the function f : ( − 1, 1) → R,f(x) = x4 uniformly continuous? Justify your
(ii) Suppose that 对 an is an absolutely convergent series. Must 对 an(3) con-
verge? Either give a proof or give a counterexample.
6. (i) Let f : R → R be the function
f(x) = {0(x)2 sin(北2(1) )
Show that f is continuous.
x 0
x = 0
(ii) Let f : [0, 1] → [2, 3] be a continuous function. Show that there exists x ∈ [0, 1] such that f(x) = x + 2.
2022-07-29