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


d.If {aₙ} = {1, ½, ⅓, ¼, ⅕, …} and
{bₙ} = {1, 0, ½, 0, ⅓, 0, ¼, 0, …},
then limₙ→∞aₙ = limₙ→∞bₙ.

Verified step by step guidance

1. Step 1: Identify the sequences given. The sequence \( \{a_n\} = \left\{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \ldots \right\} \) and the sequence \( \{b_n\} = \left\{1, 0, \frac{1}{2}, 0, \frac{1}{3}, 0, \frac{1}{4}, 0, \ldots \right\} \).
2. Step 2: Recall the definition of the limit of a sequence. A sequence \( \{x_n\} \) converges to a limit \( L \) if for every \( \epsilon > 0 \), there exists an \( N \) such that for all \( n > N \), \( |x_n - L| < \epsilon \).
3. Step 3: Analyze the limit of \( \{a_n\} \). Since \( a_n = \frac{1}{n} \), as \( n \to \infty \), \( a_n \to 0 \) because the terms get arbitrarily close to zero.
4. Step 4: Analyze the limit of \( \{b_n\} \). Notice that \( b_n \) alternates between terms of the form \( \frac{1}{k} \) (for odd indices) and 0 (for even indices). Consider the subsequences: \( b_{2k-1} = \frac{1}{k} \) and \( b_{2k} = 0 \). Both subsequences tend to 0 as \( k \to \infty \).
5. Step 5: Conclude that since both subsequences of \( \{b_n\} \) tend to 0, the entire sequence \( \{b_n\} \) converges to 0. Therefore, \( \lim_{n \to \infty} a_n = \lim_{n \to \infty} b_n = 0 \).

Key Concepts

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence approach as the index n goes to infinity. If the terms get arbitrarily close to a specific number, the sequence is said to converge to that limit. Understanding this helps determine if two sequences share the same limit.

Behavior of Subsequences

A subsequence is a sequence derived by selecting terms from the original sequence without changing their order. The limit of a sequence must be consistent with the limits of all its subsequences. If subsequences have different limits, the original sequence does not converge.

Zero Terms in a Sequence and Their Impact on Limits

Introducing zero terms intermittently in a sequence can affect its limit behavior. If zeros appear infinitely often and do not approach the original sequence's limit, the overall sequence may fail to converge or converge to a different limit. This concept is key to comparing the limits of {aₙ} and {bₙ}.