0Introduction
IA Analysis I
0 Introduction
In IA Differential Equations, we studied calculus in a non-rigorous setting. While
we did define differentiation (properly) as a limit, we did not define what a limit
was. We didn’t even have a proper definition of integral, and just mumbled
something about it being an infinite sum.
In Analysis, one of our main goals is to put calculus on a rigorous foundation.
We will provide unambiguous definitions of what it means to take a limit, a
derivative and an integral. Based on these definitions, we will prove the results
we’ve had such as the product rule, quotient rule and chain rule. We will also
rigorously prove the fundamental theorem of calculus, which states that the
integral is the inverse operation to the derivative.
However, this is not all Analysis is about. We will study all sorts of limiting
(“infinite”) processes. We can see integration as an infinite sum, and differenti-
ation as dividing two infinitesimal quantities. In Analysis, we will also study
infinite series such as 1 +
1
4
+
1
9
+
1
16
+
···
, as well as limits of infinite sequences.
Another important concept in Analysis is continuous functions. In some
sense, continuous functions are functions that preserve limit processes. While
their role in this course is just being “nice” functions to work with, they will be
given great importance when we study metric and topological spaces.
This course is continued in IB Analysis II, where we study uniform convergence
(a stronger condition for convergence), calculus of multiple variables and metric
spaces.