The history of real analysis is one of the most interesting aspects of the subject itself.

More in-depth studies of calculus were sought in order to allow the field to graduate into a form of pure mathematics. But what does this mean? How does calculus fit in with the study of all of this mathematical witchcraft? Does physics play a part at all?? Below is a brilliant article detailing the exploits made by mathematicians to accomplish what's described above. Alongside it, is an example of a collective field that made it out of the roots of real analysis, known as The Calculus of Variations:

HistoryOfRealAnalysis.pdf

https://www.britannica.com/science/calculus-of-variations-mathematics

Read it and ask yourself:

What's the purpose of mathematically proving something? Could this skill be useful for applied fields?

What is "the calculus of variations"? Why was this useful for physics, especially the field of classical mechanics?

What analysis concept effectively answers Zeno's paradox... or at least provides a more satiable mathematical structure?

Now is something a bit more fun and hands-on:

Find a ruler or some other kind of tape measure. Measure out to 3 inches and then, seek, to the best of your abilities, the number 3.1 (I'd suggest inches). Now once look at a close measurement on the ruler which is evidently greater than 3.1, like 3.2. Here's the predicament, you can look at the number pi, right? Your eyes can befall to a spot greater than 3.1 and less than 3.2. Where in the physical world is the measurement of pi located? Does pi even make sense in a non-platonic world?