Exercise · ZW-SUBSEQUENCE-009 · ★★★ Advanced
If \(n_1<n_2<\cdots\) are natural numbers, which statement always holds?
Choose one option.
Show answer
- C. \(n_k\ge k\) for every \(k\)
Explanation
By induction: \(n_1\ge1\) and \(n_{k+1}>n_k\ge k\) give \(n_{k+1}\ge k+1\). The others fail: \(n_k=k\) breaks the 2nd and 4th, and \(n_k=k^2\) breaks the 3rd.
Review the concept: Subsequence
A subsequence of a sequence \((a_n)\) is a sequence \((a_{n_k})\) made of terms of \((a_n)\) with strictly increasing natural indices \(n_k\).
\[a_{n_1},a_{n_2},a_{n_3},\dots\ \text{with}\ n_1
More practice problems: subsequence
- ZW-SUBSEQUENCE-001 Let a_n=n^2 and n_k=2k. What is the third term a_n_3 of the subsequence?
- ZW-SUBSEQUENCE-002 Which of these sequences is a subsequence of 1,tfrac12,tfrac13,tfrac14,dots?
- ZW-SUBSEQUENCE-003 Let a_n=(n+1)/(n). To what number does the subsequence a_k^2 converge?
- ZW-SUBSEQUENCE-004 Let a_n=(-1)^n. To what does the even-index subsequence a_2k converge?
- ZW-SUBSEQUENCE-005 Let a_n=3+1n. To what does the subsequence a_10^k converge?
- ZW-SUBSEQUENCE-006 Let a_n=sin ((nπ)/(2)). Which of these numbers is NOT the limit of any subsequence of (a_n)?
- ZW-SUBSEQUENCE-007 If a_2k→1 and a_2k+1→2, what happens to (a_n)?
- ZW-SUBSEQUENCE-008 Which of these index choices defines a subsequence (a_n_k) of any sequence (a_n)?