MathAnalysisCalculusLimits

Limits of sequences

Theory

Counterexamples

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A null sequence is a sequence which tends to 0. Here are some conditions which may seem to be sufficient for a sequence to be null. In fact, for each of them there is a counterexample -- a sequence satisfying this condition that isn't null. Match the conditions to the counterexamples.

Match the items from left and right columns
Each term is strictly less than its predecessor
Each term is strictly less than its predecessor while remaining positive
For any positive ε\varepsilon, the sequence has a term with absolute value smaller than ε\varepsilon
1,0,−1,−2,−3,…1, 0, -1, -2, -3, \ldots
2,32,43,54,…2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, \ldots
1,12,1,14,1,18,…1, \frac{1}{2}, 1, \frac{1}{4}, 1,\frac{1}{8}, \ldots
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