Question

PLEASE HELP!
Prove that if \[ b_n\] converges to \[ B\] and \[B \neq 0\], then there is a positive real number \[ M\] and a positive integer \[ N \] such that if \[n \geq N\], then \[\left | b_n \right |\geq M\]

Answer 1

Wait.. it converges to \(nB\)? Does \(n\) head toward infinity when we take the limit?

Answer 2

I got it fixed. It was a typo

Answer 3

Suppose \(b_n \rightarrow b>0\)

there exists a \(n_0\in N\) s.t. \(\frac{1}{n_0}<b\)

Let \(\epsilon =\frac{1}{n_0}\)

\(\exists N \ \forall \ n\ge N \ |b_n-b|<\frac{1}{n_0}\\\implies -\frac{1}{n_0}+b<b_n\)

note that \(b-\frac{1}{n_0}>0\). So now we need a case for \(b<0\)

Answer 4

Do you follow this?

Answer 5

Yes am following

Answer 6

The negative case should be similar, you can bound it away from 0, on the negative side, then look at the negative terms

Answer 7

Is there a specific theorem tied to this?

Answer 8

nope, its useful to prove many theorems.