A Convergent Sequence is Bounded: Proof, Converse

In this article, we prove that a convergent sequence is bounded. A sequence is said to be convergent if it has finite limit.

Question: Prove that A Convergent Sequence is Bounded.

Proof

Let us assume that {x_n} is a convergent sequence. So it has finite limit, say L. That is,

lim_n→∞ x_n = L.

So by definition, for every ε>0, there is a number number N such that

|x_n -L| < ε for all n≥N.

⇒ -ε < x_n -L < ε for all n≥N

⇒ L-ε < x_n < L+ε for all n≥N

Thus, x_n ∈ (L-ε, L+ε) for all n ≥ N.

Now, set

k = min {x₁, x₂, …, x_n-1, L-ε}

K = max {x₁, x₂, …, x_n-1, L+ε}.

Thus, we have shown that k ≤ x_n ≤ K for all n.

This proves that the sequence {x_n} is bounded. As a result, we conclude that every convergent sequence is bounded.

Related Study: Bounded Sequence

Converse

The converse of the theorem: “each convergent sequence is bounded” is not true.

That is, if a sequence is bounded then it may not be convergent. For example, consider the sequence {(-1)ⁿ}.

Note that {(-1)ⁿ} is a bounded sequence which is bounded by -1 and 1. But it does not converge to a definite limit as it oscillates between -1 and 1.

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