Need help clarifying limits question for Calculus. (Not really a question about calculus, more so on simplifying equ.)?

The definition said "the sequence of a converges to limit, L, provided that the terms of sequence a can be made arbitrarily close to L by taking n sufficiently large. More precisely, sequence a has the unique Limit L if given any ε > 0, it is possible to find a positive integer N (depending only on ε) such that l a - L l < ε

whenever n > N. "

The problem example confused me, not exactly the concept.

It stated that

lim n / (n - 1), as n goes to infinity, to equal 1.

It then gave that ε = 0.001 and to find the value of N that satisfies the conditions of the limit definition.

so by that definition I was given,

abs( (n / (n - 1)) - 1) < 0.01.

I added 1 to both sides and broke the fraction into two parts but... the example did things differently and I don t understand how it got there.

it converted (n / (n-1)) - 1 into 1 / (n - 1). and solved from there. Any help is appreciated.