Exercise · ZW-UNIQUENESS-OF-LIMITS-007 · ★★☆ Intermediate
With \(|L-M|=0.6\) and \(\varepsilon=0.3\), for \(n\ge N\) the triangle inequality gives \(|L-M|\le|L-a_n|+|a_n-M|<\) ...
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- D. \(0.6\)
Explanation
Each summand is less than \(0.3\), so the sum is less than \(0.6\). We get \(0.6<0.6\), absurd.
Review the concept: Uniqueness of limits
Uniqueness of limits says that a sequence \((a_n)\) cannot converge to two different numbers \(L\) and \(M\); that is why it makes sense to speak of “the” limit.
\[a_n\to L,\ a_n\to M\ \Rightarrow\ L=M\]
More practice problems: uniqueness of limits
- ZW-UNIQUENESS-OF-LIMITS-001 If a_n→2 and also a_n→ M, what is M?
- ZW-UNIQUENESS-OF-LIMITS-002 If lim _n→∞(4n+1)/(n)=L, what is L?
- ZW-UNIQUENESS-OF-LIMITS-003 To rule out a_n→3 and a_n→7 holding together, one takes ε=(|L-M|)/(2). What is it for L=3, M=7?
- ZW-UNIQUENESS-OF-LIMITS-004 In the proof, N_1 works for convergence to L and N_2 for convergence to M. If N_1=12 and N_2=30, wh…
- ZW-UNIQUENESS-OF-LIMITS-005 Which statement is true?
- ZW-UNIQUENESS-OF-LIMITS-006 If a_n→ L and a_n→ M, then a_n-a_n=0 converges to L-M and also to 0. What follows?
- ZW-UNIQUENESS-OF-LIMITS-008 An alternative proof shows |L-M|<ε for every ε>0. Which fact lets us conclude L=M?
- ZW-UNIQUENESS-OF-LIMITS-009 If (-1)^n→ L, with ε=tfrac12 the terms 1 and -1 (which appear for large n) satisfy |1-L|<tfrac12 an…