Mathematics 2C - Introduction to Real Analysis 2020
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Mathematics 2C - Introduction to Real Analysis
2020
1. (i) For each of the following sets, state whether the infimum exists and whether the supremum exists, and give their values when they do. You do not need to justify your answers.
(a) X = [-2, 4) u (-3, 2) u [3, 7].
(b) Y =I 会
4 + 2 | x < -4、.
(c) Z = {x e R | (x + 1)2 < 2}.
(ii) Let A be a set which has an infimum and supremum, with inf(A) > 0. Define the set B by:
B = {a2 + 2a + 
| a e A} .
Prove that sup(B) exists.
2. (i) Prove directly from the definition that the following sequence converges: x。 = 
+ 3.
(ii) Calculate the limit
。(l)
╱ 
- n、,
stating clearly all properties of limits used.
(iii) Prove directly from the definition that the following sequence does not con-
|
verge to |
any |
limit:
|
if n is divisible by 3; otherwise. |
3. (i) Prove that the sequence x。 = (2。 + 3。) ←/。 is convergent.
(ii) Suppose (x。)
is a sequence such that 会(会)
→ 0 as n → o. Show that
x。 → 2 as n → o.
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)
。车← n!
(ii) & 2 3。 - cos(2n) 。车←
(iii) & n3 - n 3n4 + 3n2 + 2 。车←
5. (i) Give an example of a conditionally convergent series with sum zero. Prove that your series is convergent.
(ii) Suppose that Either give a proof or give a counterexample.
6. (i) Prove, directly from the definition of continuity, that the function f : R → R given by f (x) = 4x - 7 is continuous.
(ii) Let f : R → R be the function
f (x) = ,0(x) sin( Show that f is continuous at 0 e R.
(iii) Let f : [-1, 1] → [0, 2] be a continuous function. Show that there exists a point x e [-1, 1] for which f (x) = 1 - x. |
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2022-07-29