## 3.1. Sequences

### Proposition 3.1.4: Convergent Sequences are Bounded

### Proof:

Let's prove uniqueness first. Suppose the sequence has two limits,*a*and

*a'*. Take any

*> 0*. Then there is an integer

*N*such that:

if| a_{j}- a | <

*j > N*. Also, there is another integer

*N'*such that

if| a_{j}- a' | <

*j > N'*. Then, by the triangle inequality:

if| a - a' | = | a - a_{j}+ a_{j}- a' |

|a_{j}- a | + | a_{j}- a' |

< + = 2

*j > max{N,N'}*. Hence

*| a - a' | < 2*for any

*> 0*. But that implies that

*a = a'*, so that the limit is indeed unique.

Next, we prove boundedness. Since the sequence converges, we can take, for
example, * = 1*. Then

if| a_{j}- a | < 1

*j > N*. Fix that number

*N*. We have that

for all| a_{j}| | a_{j}- a | + | a | < 1 + |a|

*j > N*. Define

ThenM = max{|a_{1}|, |a_{2}|, ...., |a_{N}|, (1 + |a|)}

*| a*for all

_{j}| < M*j*, i.e. the sequence is bounded as required.