/sci/ - Science & Math

>>14812817
>the rules of using limits? what are (all of) them?
What do you mean? Do you mean specifically: what techniques are admissible when proving things about the real numbers using Cauchy sequences? I don’t know of any such explicit list. But here is an example using the definitions in >>14812561
>Thm: There is no infinitesimal real number, i.e. if x is a real number then either x=0 or its Cauchy sequence eventually gets stuck in an interval [q_1,q_2] that doesn’t have zero in it.
>Pf: Let c be the Cauchy sequence and suppose it never gets stuck in an interval that doesn’t have zero in it. Now let z be the Cauchy sequence consisting of just 0s and consider the sequence c’ we get by alternating entries of c and z. For any given size s we know that c eventually gets stuck in an interval of size <s; and, by our starting assumption, this interval necessarily has 0 in it. Therefore (trivially) z gets stuck in the interval too and hence so does c’. Therefore c’ is Cauchy. Therefore x=0 by the definition of equality for real numbers.

>>14812817
>the rules of using limits? what are (all of) them?
What do you mean? Do you mean specifically: what techniques are admissible when proving things about the real numbers using Cauchy sequences? I don’t know of any such explicit list. But here is an example using the definitions in >>14812561
>Thm: There is no infinitesimal real number, i.e. if x is a real number then either x=0 or its Cauchy sequence eventually gets stuck in an interval [q_1,q_2] that doesn’t have zero in it.
>Pf: Let c_x be the Cauchy sequence and suppose it never gets stuck in an interval that doesn’t have zero in it. Then the intervals of smaller and smaller size always have zero in them. Now let c_0 be the Cauchy sequence consisting of just 0s and consider the sequence c’ we get by alternating c_x and c_0. For any given size s we know that c_x eventually gets stuck in an interval of size <s; and by the discussion above, this interval necessarily has 0 in it. Therefore c’ gets stuck in the interval too. Therefore c’ is Cauchy. Therefore x=0 by the definition of equality for real numbers.