Mathematical analysis according to Fichtenholtz?

Question

Mathematics

in the first chapter Fichteholtz proves that the limit of Xn = (n^2-n+2)/(3n^2+2n-4) is 1/3.
and he writes "for this purpose, consider the difference Xn-1/3 =(5n-10)/(3(3n^2+2n-4)) and estimate its absolute value; for n>2 we have: |Xn-1/ 3|=(5n-10)/(3(3n^2+2n-4))<5n/(3(3n^2-4))<5n/(3*2n^2)<1/n"
as he did it

Answer 1

Well, look, you can tend to zero from below, from above, or you can take turns. For example (-1)^2 / n.
This means that you can consider the module, and the result of striving will not change.
So, in your example, the modulus of the subtracted limit sequence is always less than 1/n, which in turn tends to zero. This means that Xn - 1\3 tends to zero, and Xn - to 1\3, which is what tr.doc.