Limits and subsequences If the terms of one sequence appear...

Limits and subsequences If the terms of one sequence appear in another sequence in their given order, we call the first sequence a subsequence of the second. Prove that if two sub-sequences of a sequence $\left\{a_{n}\right\}$ have different limits $L_{1} \neq L_{2},$ then $\left\{a_{n}\right\}$ diverges.

Limits and subsequences If the terms of one sequence appear
in another sequence in their given order, we call the first sequence
a subsequence of the second. Prove that if two sub-sequences of
a sequence $\left\{a_{n}\right\}$ have different limits $L_{1} \neq L_{2},$ then $\left\{a_{n}\right\}$ diverges.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Divergence of a Sequence

A sequence is said to diverge if it does not converge to a unique limit. The presence of subsequences converging to different limits is a direct indication that the original sequence does not settle on a single value, thereby diverging.

Uniqueness of Limits in Convergence

A key property in the study of sequences is that if a sequence converges, every subsequence must converge to the same limit. This uniqueness is essential when assessing the overall behavior of the sequence.

Subsequence

A subsequence is a sequence derived from another sequence by selecting elements in the original order while possibly omitting some elements. It maintains the relative order of appearance, which is crucial for studying the behavior of parts of the original sequence.

Limit of a Sequence

The limit of a sequence is the value that the sequence approaches as the index goes to infinity. This concept formalizes the idea of convergence, where the sequence’s terms get arbitrarily close to a single value.

Transcript

Please give Ace some feedback