Exercise · ZW-UNIQUENESS-OF-LIMITS-008 · ★★★ Advanced
An alternative proof shows \(|L-M|<\varepsilon\) for every \(\varepsilon>0\). Which fact lets us conclude \(L=M\)?
Choose one option.
A A nonnegative real smaller than every \(\varepsilon>0\) is zero B \(L\) and \(M\) are rational C The sequence is monotone D \(\varepsilon\) may be taken negative Show answer A. A nonnegative real smaller than every \(\varepsilon>0\) is zero← Back to the question
Explanation If \(|L-M|>0\), taking \(\varepsilon=|L-M|/2\) contradicts the inequality; hence \(|L-M|=0\).
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