We can easily do any problems related to it by working out these problems.

Course Material 2.2 Sequences and Their Convergence (2)

Definition – Converging Sequences

A real sequence 𝑥𝑛 is said to converge to a real number 𝑥 if for any given 𝜖 > 0,
there exists a positive integer 𝑁 such that for all 𝑘 ≥ 𝑁 w have 𝑥𝑘 ∈ 𝑥 − 𝜖, 𝑥 + 𝜖 that
is, for all 𝑘 ≥ 𝑁, 𝑥𝑘 − 𝑥 < 𝜖.

In this case, we say that 𝑥𝑛 is a convergent sequence, converging to the limit 𝒙 and
we may write lim𝑛 →∞ 𝑥𝑛 = 𝑥 or simly 𝑥𝑛 → 𝑥.

If 𝑥𝑛 does not converge to any real number, we say that it is a divergent sequence.

Note:

We may note from the definition that the convergence and the limit of the sequence do not
depend on the first finitly many terms of the sequence. They depend only on the terms 𝑥𝑛
for all 𝑛 ≥ 𝑁 for some integer 𝑁 which may be taken as large as is needed. This part of the
sequence is usually called a Tail of the sequence.

Examples

1. The constant sequence 𝑐 is convergent and converges to 𝑐.
Proof: We have 𝑥𝑘 = 𝑐 for all 𝑘 ∈ ℕ so that 𝑥𝑘 − 𝑐 = 𝑐 − 𝑐 = 0 < 𝜖 for all 𝑘 ≥ 1
1
2. Let 𝑥𝑛 = 𝑛 for all 𝑛 ∈ ℕ then 𝑥𝑛 converges to 0
1 1
Proof: We have 𝑥𝑛 = 𝑛 for all 𝑛 ∈ ℕ so that 𝑥𝑛 − 0 = 𝑛
1
For any 𝜖 > 0, we have > 0 ,then by Archimedes theorem, there is a positive
𝜖
1 1 1 1
integer 𝑁 such that 𝑁 > 𝜖 so that 𝑁 < 𝜖. But then for all 𝑛 ≥ 𝑁 we have 𝑛 ≤ 𝑁 < 𝜖
1
Hence it follows that for all ≥ 𝑁 𝑥𝑛 − 𝑥 < 𝜖. Or converges to 0
𝑛
1
3. Let 𝑥𝑛 = 2𝑛 for all 𝑛 ∈ ℕ then 𝑥𝑛 converges to 0
1 1 1 1
Proof: In this case as 2𝑛 < 𝑛 , as in the above case 𝑥𝑛 − 0 = 2𝑛 < 𝑛 < 𝜖 for all 𝑛 ≥ 𝑁
−1 𝑛 +1
4. Let 𝑥𝑛 = 𝑛
for all 𝑛 ∈ ℕ then 𝑥𝑛 converges to 0
1
Proof: This is similar to the case for 𝑥𝑛 = 𝑛