Questions
1. *
Let x and y be positive real numbers with x < y. Prove that x2 < y2.
2. *
Prove that if a < b and c < d then a + c < b + d.
3. *
 1 
Prove that the sequence   tends to zero.
 n
4. *
(a) Prove that if
1
= 0
n
n a n
(b) Give a counterexample to show that the converse (if lim a n = 0 then
lim a n =  then lim
n
1
= ) is false.
lim
n a n
5. * Give an example of a sequence {an} satisfying all of the following:
{an} is monotonic
0 < an < 1 for all n and no two terms are equal
1
lim a n =
2
n
6. * Let k > 0 be a constant and consider the important sequence {kn}. It’s behaviour
as n   will depend on the value of k.
(i) State the behaviour of the sequence as n   when k = 1 and when k = 0.
(ii) Prove that if k > 1 then kn   as n  
(hint: let k = 1 + t where t > 0 and use the fact that (1 + t)n > 1 + nt.
(iii) Prove that if 0 < k < 1 then kn  0 as n   .
7. * Given a geometric series with first term a > 0 and common ration r > 0 prove that
a finite “sum to infinity” exists if and only if r < 1 and show that in this case the
a
sum to infinity is
.
1 r
(hint: use the results of Q6 above, and recall that the sum of the first n
a (1  r n )
terms is
).
1 r
8. * Use the definition of f(x)   as x   to show that f(x) = x + 2   as x  
.
9. *
(a) Show that f(x) =
3
tends to  as x  –2+ .
x2
1
(b) Show that f(x) =
1
tends to  as x  4–
4x
4 when x  5
10. * Let f(x) = 
.
0 when x  5
Show that f is not continuous at x = 5.
11. * Prove that if f is continuous at all values of x then so is kf where k is a constant.
12. * Give an example of a function that is continuous at all non-integer values but is
discontinuous at all integer values. Prove the discontinuity property of your
function (i.e. that it is discontinuous at all integer values).
13. * Explain why this is no good as a definition of continuity at a point a (either by
giving an example of a continuous function that does not satisfy the definition or a
discontinuous one that does):
Given  > 0 there exists a  > 0 such that |x – a| <   |f(x) – f(a)| < 
14. * Can a function be continuous at one value of x and discontinuous at all other
x  R? Explain your answer, giving proofs where appropriate.
15. * Show f(x) =
2
is not uniformly continuous on the interval (0, 1).
x
.
16. * Prove that if f and g are functions that are uniformly continuous on an interval I
then so is f + g. (Hint: a proof very similar to that for Example 3 will work, you
just have to adapt it to the definition of uniform continuity).
1
1
17. * Show that if f(x) =
then f (c) = 
.
3
x
2 c
1  x 2
18. * Show that the function f(x) = 
 1  x
when x  0
is not differentiable at x = 0.
when x  0
19. * What can you say about the differentiability of fg at x = c in each of the
following cases? (Here the function fg is defined by fg(x) = f(x)g(x)).
(a) f is differentiable at c, but g is not.
(b) f is not differentiable at c, and neither is g.
(c) both f and g are differentiable at c.
2
20. * Let f(x) = 5 x . Show that for c  0, f is differentiable at c, with f (c) =
1
5
5 c4
,
but that f is not differentiable at 0.
(You will find the identity a5 – b5 = (a – b)(a4 + a3b + a2b2 + ab3 + b4) useful).
21. * Prove that between any two roots of x4 – 4x2 + k = 0 there is at least one root of
x3 = 2x.
22. * Suppose f is continuous and differentiable on [0, 1], that 0  f(x))  1 for each x in
[0, 1], and that f (x)  1 each x in [0, 1]. Show that there is at most one solution of
f(x) = x in [0, 1].
23. * In the Cauchy Mean Value Theorem take f(x) = x2, g(x) = 4x3 – 3x4 on the interval
[0, 1]. Show that the conditions of the theorem hold and find possible values of
"c".
3