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#1
How do i do these questions ive no clue.

A. Suppose that an < 0 ∀ n ∈ N+ and that limn→∞ an = 0. Prove that the sequence ( 1/an )n∈N+ is unbounded below.

Prove true or false

(A) ∀ δ > 0, ∃ M ∈ N+ such that n > 2M ⇒ |an − L| << δ.
(B) ∀ δ > 0, ∃ M ∈ N+ such that n > M ⇒ |an − L| << δ.
(C) ∀ δ > 0, ∃ M ∈ N+ and C > 0, where M and C may depend on δ and the sequence (an)n∈N+ , such that n > M ⇒ |an − L| < Cδ.
(D) ∀ δ > 0, ∃ M ∈ N+ and C > 0, where C is independent of δ and the sequence (an)n∈N+ , such that n > M ⇒ |an − L| << Cδ; M may depend on δ and (an)n∈N+ .

δ = delta
<< means less than or equal to
0
3 years ago
#2
For the first one as the limit of the sequence is 0, we know that for any number ε=1/M (M>0) we can find an N such that for large enough n. What can you say about and -M?
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