Date: Oct 4, 2017 2:40 PM
Author: Markus Klyver
Subject: No John Gabriel, an infinite sum is a limit and 0.333... is indeed<br> 1/3. Plus several other misconceptions Gabriel has

We define the sum ? (a_i) from i=0 to ? as the limit of ? (a_i) from i=0 to n as n approaches ?. Under this definition 0.333... is indeed 1/3.  This is a DEFINITION of a very useful short-hand notation.

a_1 + a_2 + a_3 + ... is an infinite sum, and defined AS A LIMIT.

a_1 + a_2 + a_3 + ... is not the same as a_1 + a_2 + a_3 + ... + a_n and a_1 + a_2 + a_3 + ... is not "a very long but finite sum".

a_1 + a_2 + a_3 + ... is an infinite sum, which is DEFINED to be lim_{n --> ?} (a_1 + a_2 + a_3 + ... + a_n). All infinite sums are defined as limits of finite sums, because you can't really add an "infinite amount of terms". Which is why we DEFINE INFINITE SUMS AS LIMITS.

Also, sequences can be constant and a sequence is usually defined as a function from the natural numbers into the set we're interested in. And no, not all Cauchy sequences are convergent. It depends on the structure you are considering. And no, nothing in the limit definition requires us to know what the limit is. We can perfectly define limits without knowing how to prove limits or evaluating limits.

And no, we don't define real numbers as limits of Cauchy sequences. That's an other strawman and completely illogical. We can, however, define a real number as an equivalence class of rational Cauchy sequences. An equivalence class is not a limit. An equivalence class is a set of equivalent elements under a certain equivalence relation.