Mathematician KP Hart Answers Math Questions for Us and It's Awesome!

Mathematician and GGA friend KP Hart lives in the Netherlands. Recently, the Dutch government asked its population to post questions of scientific interest through The Dutch Science Agenda. Hart went looking through the questions and noticed some questions already had answers so he created a tumblr page to address the questions. He writes on his page:

In the beginning of this year the Dutch government opened a website, The Dutch Science Agenda, where everyone could post questions that they thought were of scientific interest. This was an attempt to involve the whole country in determining what the Dutch science agenda should be in the coming years.
I looked through the questions and searched for terms like `mathematics’, `infinity’ … to see what mathematical questions there were and I noticed various questions that already have answers (and have had for a long time). On a whim I decided to post answers to those questions, in Dutch. For your edification I will translate these posts into English.

We thought this was fabulous and reached out to Hart and he’s humbly let us post his answers. We’ll be posting a couple questions a week! Enjoy!

First question: Is 0.99999etc (so, with infinitely many nines after the decimal point) equal to 1?

The answer to this question depends of course on what you think 0.99999… should mean. In Mathematical Analysis this has an unambiguous meaning. To begin: “infinitely many nines” means “a sequence of nines indexed by the natural numbers”. Next: this sequence of nines determines a sequence of real numbers, for every natural number n take the first n nines and calculate the sum 9/10+9/100+…+9/10ⁿ (this is how we interpret finite decimal expansions), the result is equal to 1-1/10ⁿ. The real number that corresponds to 0.99999… is the smallest one that is larger than than all these values. These values are all less than 1, so that real number is at most 1. On the other hand if r is less than 1 then there is an n such that r is less than 1-1/10ⁿ, so all numbers below are out. Thus the only reasonable value we can assign to 0,99999… is 1.

The expression 0.999… has its own Wikipedia page where you can find much more info and other reasons why its value should be 1.