The epsilon definition of a limit
Objective. State and use the formal definition of sequence convergence.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
aₙ → L means: for every ε > 0 there is N such that |aₙ − L| < ε for all n > N. Smaller ε simply forces a larger N.
Worked example
Show 1/n → 0 using the definition.
- 1|1/n − 0| = 1/n.
- 2We need 1/n < ε, i.e. n > 1/ε.
- 3So take N = ⌈1/ε⌉.
Answer: For any ε > 0, N = ⌈1/ε⌉ works
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Practice
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Start practisingCommon mistake. Choosing N before ε, reversing the order of the quantifiers.
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