1=0.999…. A Formal Proof (Kind of)

In this article, I present a formal proof that the recurring decimal 0.999…. is in fact equal to 1. In doing so, however, I will sidestep a little of the more formal notation and strictness that a truly airtight proof would require. Instead, I will opt for presenting the main concepts required for the proof and show the construction of the proof based off these.

With this in mind, before we can begin the construction of our proof, we require the use of some techniques and ideas from a branch of mathematics known as real analysis. In particular, we first need to discuss the concepts of a sequence and a limit.

What is a sequence?

Simply put, a sequence is a list of elements in which order matters. We generally denote a sequence using curly brackets, { }. So, for example, the sequences {1, 2, 3} and {1, 3, 2} are considered different, even though they contain the same elements. There are two types of sequences:
(i) Finite sequences, which simply means you have a finite number of elements in the sequence, such as the sequences {1, 2, 3}and {1, 3, 2}.
(ii) Infinite sequences, which simply means you have an infinite number of elements in the sequence, such as the sequence of natural numbers, ℕ = {1, 2, 3, 4,….}.

It should be clear that when it comes to writing down an infinite sequence it is not possible to write down all the terms of the sequence. So, if we cannot write down all the terms, how do we tell someone what a sequence looks like? We give them a rule.

Above, when I used the natural numbers as an example of an infinite sequence, I only gave part of the sequence, {1, 2, 3, 4,….}, but you didn’t need me to tell you how to continue this. You already knew the rule, you’ve known it basically your whole life, although you probably no longer think about it.

To write down the sequence of natural numbers you start at 1 and then add 1 to get the second element, 2. You then start at the second element and you add 1 to get the third element, 3. You continue to repeat this process to obtain all the natural numbers.

This is a simple set of instructs you could give someone for constructing the sequence of natural numbers. Unfortunately, it isn’t always so easy to write down such a set of instructions. This is why we have a more formal, and concise, way of describing a sequence.