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uva123
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Homework Statement
Prove if {bn} converges to B and B ≠ 0 and bn ≠ 0 for all n, then there is M>0 such that |bn|≥M for for all n.
Homework Equations
What I have so far:
I know that if {bn} converges to B and B ≠ 0 then their is a positive real number M and a positive integer N such that if n≥N, then |bn|≥M . (by lemma)
PROOF (of lemma)- (Note: let E be epsilon)
since B ≠ 0, (|B|)/2=E>0. There is N such that if n≥N, then
|bn-B|<E. Let M=[(|B|)/2]. thus for n≥N,
|bn|=|bn-B+B|≥|B|-|bn-B|≥|B|-[(|B|)/2]=[(|B|)/2]=M
The Attempt at a Solution
i know that this is not the entire proof i need but i don't know what changes need to be made. what variations do i need to make in the proof for the lemma? please help point me in the right direction!