Convergent Sequences are Bounded

Let

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

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:

| a_j - a | < if j > N

Also, there is another integer N' such that

| a_j - a' | < , if j > N'

Then, by the triangle inequality:

| a - a' | < | a_j - a | + | a_j - a' | < + = 2 if j > max{N,N'}.

Hence | a - a' | < 2 for any > 0. But then a = a', so that the limit is indeed unique.

Next, we prove boundedness. Since the sequence converges, we can take any = 1. Then

| a_j - a | < 1 if j > N

Fix that number N. Then we have that

| a_j | < = | a_j - a | + | a | < 1 + |a| := P for all j > N

But then define M = max{| a_j |, j = 1,.., N, P}. Then | a_j | < M for all j, i.e. the sequence is indeed bounded.

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