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…