Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.

c.The convergent sequences {aₙ} and {bₙ} differ in their first 100 terms, but aₙ = bₙ for n > 100.
It follows that limₙ→∞aₙ = limₙ→∞bₙ.

검증된 단계별 안내

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

Note that the behavior of a sequence at finitely many terms (such as the first 100 terms) does not affect the limit, because the limit depends only on the terms as \( n \to \infty \).

Given that \( a_n = b_n \) for all \( n > 100 \), the two sequences are identical from the 101st term onward.

Since both sequences \( \{a_n\} \) and \( \{b_n\} \) are convergent, and they agree on all terms beyond \( n = 100 \), their tails are the same and must converge to the same limit.

Therefore, it follows that \( \lim_{n \to \infty} a_n = \lim_{n \to \infty} b_n \), regardless of differences in the first 100 terms.

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Definition of Convergent Sequences

A sequence {aₙ} converges to a limit L if, for every positive ε, there exists an N such that for all n > N, |aₙ - L| < ε. This means the terms get arbitrarily close to L as n becomes large, regardless of the initial terms.

Effect of Finite Differences on Limits

Changing a finite number of terms in a sequence does not affect its limit. Since limits depend on the behavior of terms as n approaches infinity, differences in the first 100 terms do not alter the limit of the sequence.

Uniqueness of Limits for Sequences

If two sequences are equal from some index onward and both converge, they must share the same limit. This follows because their tails coincide, so their limiting behavior is identical.

Explain why or why notDetermine whether the following statements are true and give an explanation or counterexample.c.The convergent sequences {aₙ} and {bₙ} differ in their first 100 terms, but aₙ = bₙ for n > 100.It follows that limₙ→∞aₙ = limₙ→∞bₙ.