Question on Negation of Definition of limit of Sequence

Question

Press/click below hints, answer "buttons" for solution: 

Do you really need hint? Give your 100% first! If still you have problem then check hint 2

Hint 1

"given that $\lim_{x \to +\infty} a_{n}= 0$. Hence by definition of limit of sequence we have,

for given any $\epsilon >0$ there exists $K\in\mathbb{N}$ such that, $|a_n-0|<\epsilon$

i.e. given any $\epsilon >0$ there exists $K\in\mathbb{N}$ such that, $|a_n|<\epsilon$

"The question ask negation of above statement"

Which is given by "there exists an $\epsilon>0$ such that for every $K\in\mathbb{N}$ there exists $n>K$ such that $|a_n|\geq\epsilon$ from this can you conclude the required answer ?

Hint 2

By Hint2 we conclude that the answer is option (a)


Can you discard the other options? yes! Find counterexample for discarding other options. Did you find any? ("I will not do spoon feeding" If You have to learn something then you need to work by yourself)

Task: You can comment counterexamples to discard options (b), (c), (d) in below comment box. I will verify them and reply you.

Answer



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