# Interactive Real Analysis

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## 3.1. Sequences

### Proposition 3.1.4: Convergent Sequences are Bounded

Let be a convergent sequence. Then the sequence is bounded, and the limit is unique.

### 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:
| aj - a | < if j > N. Also, there is another integer N' such that
| aj - a' | < if j > N'. Then, by the triangle inequality:
| a - a' | = | a - aj + aj - a' | |aj - a | + | aj - a' |
< + = 2 if 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

| aj - a | < 1
if j > N. Fix that number N. We have that
| aj | | aj - a | + | a | < 1 + |a|
for all j > N. Define
M = max{|a1|, |a2|, ...., |aN|, (1 + |a|)}
Then | aj | < M for all j, i.e. the sequence is bounded as required. Next | Previous | Glossary | Map