Is it true that a sequence {aₙ} of positive numbers must converge if it is bounded above? Give reasons for your answer.

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Recall the definition of convergence for a sequence: a sequence \( \{a_n\} \) converges if there exists a limit \( L \) such that for every \( \epsilon > 0 \), there is an \( N \) where for all \( n > N \), \( |a_n - L| < \epsilon \).

Understand that being bounded above means there exists some number \( M \) such that \( a_n \leq M \) for all \( n \), but this does not guarantee the sequence approaches a specific value.

Consider that a sequence can be bounded above but still oscillate or fail to settle down to a single limit, for example, a sequence that jumps between two values less than \( M \).

Note that for convergence, the sequence must get arbitrarily close to one particular number, not just stay below a certain bound.

Conclude that a sequence of positive numbers bounded above does not necessarily converge; boundedness alone is not sufficient to ensure convergence.

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주요 개념

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Bounded Sequences

A sequence is bounded above if there exists a number that is greater than or equal to every term in the sequence. Being bounded means the terms do not grow beyond a certain limit, but it does not guarantee convergence by itself.

Convergence of Sequences

A sequence converges if its terms approach a specific finite limit as the index goes to infinity. Convergence requires the terms to get arbitrarily close to a single value, not just to be bounded.

Monotone Convergence Theorem

This theorem states that every bounded monotone sequence converges. If a sequence is both bounded and monotone (either non-increasing or non-decreasing), it must converge, but boundedness alone is insufficient.