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tmua.fyi Paper 2 Challenge 2 · Q9

Paper 2Logic & proofSequences & series

A sequence (an)(a_n) and a number LL are given. Consider the statement

for every ε>0 there exists N such that ∣an−L∣<ε for all n>N.\text{for every }\varepsilon>0\text{ there exists }N\text{ such that }|a_n-L|<\varepsilon\text{ for all }n>N.

Which of the following is its negation?

AFor every ε>0\varepsilon>0 there exists NN such that ∣an−L∣≥ε|a_n-L|\ge\varepsilon for all n>Nn>N.
BThere exists ε>0\varepsilon>0 such that for every NN there exists n>Nn>N with ∣an−L∣≥ε|a_n-L|\ge\varepsilon.
CThere exist ε>0\varepsilon>0 and NN such that ∣an−L∣≥ε|a_n-L|\ge\varepsilon for all n>Nn>N.
DFor every ε>0\varepsilon>0 and every NN there exists n>Nn>N with ∣an−L∣≥ε|a_n-L|\ge\varepsilon.
EThere exists ε>0\varepsilon>0 such that ∣an−L∣≥ε|a_n-L|\ge\varepsilon for every nn.
FFor every NN there exist ε>0\varepsilon>0 and n>Nn>N with ∣an−L∣≥ε|a_n-L|\ge\varepsilon.
GThere exists NN such that for every ε>0\varepsilon>0 there exists n>Nn>N with ∣an−L∣≥ε|a_n-L|\ge\varepsilon.
HFor every ε>0\varepsilon>0 there exists NN such that ∣an−L∣>ε|a_n-L|>\varepsilon for some n>Nn>N.

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