1. When working with limits it is important to prove the limit exists before starting to use the
algebraic properties. The following example is intended to illustrate what can go wrong.
• A ’proof ’ of the equality 0 = 1.
Consider the sequences an = sin(n) and bn = cos(n). We know sin(n + 1) = sin(n) cos 1 +
cos(n) sin 1, and cos(n + 1) = cos(n) cos 1 − sin(n) sin 1. Hence by shift rule and algebra of
limits
lim an = lim an sin 1 + lim bn cos 1
lim bn = lim bn cos 1 − lim an sin 1
By solving the linear system we find lim an = lim bn = 0.
Moreover, since sin2 (n) + cos2 (n) = 1, then
lim a2n + lim b2n = 1
Therefore 0 = 1.
2. When proving that a sequence is contracting it is important to check that λ < 1 works
uniformly, that is, it does not depend on n. In other words, the assumption of the theorem
is sharp, for example, the following claim is false.
• Let (an ) be a sequence satisfying |an+2 − an+1 | ≤ λn |an+1 − an |, where 0 < λn < 1,
for all n. Then (an ) is a converging sequence.
Counterexample. Let an =
Pn
1
k=1 k .
We can see
1
n+2
n+1
1
=
·
n+2 n+1
n+1
=
|an+1 − an |
n+2
|an+2 − an+1 | =
Hence, this sequence satisfies the hypothesis with λn =
However we know that lim an = ∞.
1
n+1
.
n+2