Exercise · ZW-UNIQUENESS-OF-LIMITS-010 · ★★★ Advanced
A student says: "\(a_n\to L\) and \(a_n\to M\) with \(L\ne M\) is not contradictory because \(\varepsilon\) can be chosen large". Where is the error?
Choose one option.
A The definition only requires one fixed \(\varepsilon\) B The definition requires the condition for every \(\varepsilon>0\), and a small \(\varepsilon\) breaks it C It is correct if \(L\) and \(M\) are close D It is correct if \(N\) is large Show answer B. The definition requires the condition for every \(\varepsilon>0\), and a small \(\varepsilon\) breaks it← Back to the question
Explanation Convergence holds for every \(\varepsilon>0\); with \(\varepsilon\le|L-M|/2\) the two conditions are incompatible.
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