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Having a numpty moment here with a proof, so any help would be great! Thanks!

If (xn) is a convergent sequence with a non-zero limit, prove that there exists a positive integer N such that mod(xn)>amod(limxn) where n tends to infinity for the lim part there and 0<a<1

So I let the limxn bit just be l to make life a bit easrer (hope it's okay to do that) and I statred trying to prove things using the triangle inequlity etc, but it just won't seem to work. For example-

mod(xn)=mod(xn - l + l)<mod(xn-l) + mod(l). but I couldn't get anything from this.

or mod(xn)=mod(xn -l +l)> mod(mod(xn+l) - mod(l))=mod(mod(l)- mod(xn+l))> mod(l)- mod(xn+l) > amod(l)-mod(xn+l) However, I can't get anything from this either. Slightly bamboozled here which is a bit worrying cosnidering it's not exactly rocket scienece being required to get it.

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# Trouble with a converging sequence.

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