Analysis is different from earlier mathematics such as calculus. And we need to learn new knowledge and skills. Let’s take a look at a typical page in an analysis book.
The left side box shows a typical page of analysis books. At first glance, this is frightening and boring. But it’s actually very exciting since we are going to learn some sophisticated mathematics.
This page contains many symbols and abbreviations. The right side box shows what each means and how to pronounce it. These symbols work as abbreviations such that we can express complex mathematical ideas in a condensed form. We should master them before doing anything else. They are very, very powerful.
‘a n’ is a general sequence. The right arrow means ‘tends to’ or ‘converges to’. This upside-down A means ‘for all’ or ‘for every’. This is the Greek letter epsilon, it is used as a variable. This reversed E means ‘there exists’ or ‘this is, there are’. This set membership symbol is read as ‘in’ or ‘is an element of’. This big N represents the natural numbers set, which includes 1, 2, 3, and so on. ‘max’ means ‘the maximum of’. And lastly, the curly brackets represents a set, here it is the set that contains the numbers N1 and N2.
Let’s look at the lecture note and identify its key components. It contains a definition, a theorem and a proof. The definition tells us what it means for a sequence to converge to a limit. The theorem tells what happens when we combine two convergent sequences ‘a n’ and ‘b n’ by multiplying their respective terms. The proof is an logical argument that concludes that the theorem is true. The proof first assumes that the two sequences ‘a n’ and ‘b n’ satisfy the definition, and then concludes that the multiplied sequence ‘a n times b n’ satisfies the definition too, thus proves the theorem.
The note might look like a jumble of symbols and words, but it can read aloud like any other sentences. Let’s try it.
Definition: A sequence an converges to a if and only if that for every number epsilon bigger than zero, there exists a natural number N such that for every number n greater than big N, the absolute value of ‘a n’ minus a is less than epsilon.
Theorem: Suppose that a sequence ‘a n’ converges to ‘a’, and another sequence ‘b n’ converges to ‘b’, then the sequence of the ‘a n times b n’ converges to ‘a times b’. The sequence ‘a n times b n’ is made by multiplying the corresponding ‘a n’ and ‘b n’ terms.
Proof: Let a sequence ‘a n’ converge to ‘a’, and the sequence ‘b n’ converge to b.
Let a variable epsilon greater than zero be arbitrary.
Then there exists a natural number ‘N 1’ such that for every ‘n’ greater than ‘N 1’, the absolute value of ‘a n minus a’ is less than epsilon over ‘2 times the absolute value of b plus 1’.
Also sequence ‘a n’ is bounded since all convergent sequences are bounded.
So there exists a number ‘M’ greater than zero, such that for all ‘n’ in the Natural Number set N, the absolute value of ‘a n’ is less or equal to ‘M’
For this ‘M’, there exits a number ‘N 2’ in the natural number set N, such that for all ‘n’ greater than ‘N 2’, the absolute value of ‘b n minus b’ is less than epsilon over ‘2 M’.
Let N be the maximum of the set ‘N 1 and N 2’.
Then for all ‘n’ greater than ‘N’, the absolute value of ‘a n times b n minus a times b’ is equal to the absolute value of ‘a n times b n minus a n times b plus a n times b minus a times b’.
Which is less than or equal to the absolute value of ‘a n times b n minus b’ plus the absolute value of b times a n minus a.
Which equals to the absolute value of ‘a n’ times the absolute value of ‘b n minus b’ plus the absolute value of ‘b’ times the absolute value of ‘a n minus a’.
Which is less than ‘M times epsilon over 2 M’ plus ‘the absolute value of b times epsilon over 2 times the absolute value of b plus 1’.
Which is less than epsilon over 2 plus epsilon over 2
Which equals epsilon.
Hence the sequence of ‘a n times b n’ converges to a times b.
Analysis is pure mathematics and should be understood as a theory, a network of general results liked together by valid logical arguments known as proofs. Analysis does not focus on repetitious calculations like calculus, but rather it focuses on the theory. It is the theorems, proofs and ways of thinking about them that you need to understand and master.