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Is convergent sequence always bounded?

  1. Is convergent sequence always bounded?
  2. Can a sequence converge and not be bounded?
  3. Is every sequence bounded?

Is convergent sequence always bounded?

Theorem 1 (Theorem 9.1). Every convergent sequence is bounded.

Can a sequence converge and not be bounded?

If a sequence is not bounded, it is an unbounded sequence. For example, the sequence 1/n is bounded above because 1/n≤1 for all positive integers n. It is also bounded below because 1/n≥0 for all positive integers n. Therefore, 1/n is a bounded sequence.

Is every sequence bounded?

Proof: Every sequence in a closed and bounded subset is bounded, so it has a convergent subsequence, which converges to a point in the set because the set is closed.

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