We can use mathematical induction which is deduced from Peano axioms and illustrated on Terence Tao's Real Analysis(here it is)
- Axiom 2.1 $0$ is a natural number.
- Axiom 2.2 If $n$ is a natural number, then $n++$ is also a natural number.
- Axiom 2.3 $0$ is not the successor of any natural number; i.e., we have $n++\neq 0$ for every natural number $n$.
- Axiom 2.4 Different natural numbers must have different successors; i.e., if $n,m$ are natural numbers and $n\neq m$, then $n++\neq m++$. Equivalently, if $n++=m++$, then we must have $n=m$.
- Axiom 2.5 (Principle of mathematical induction). Let $P(n)$ be any property pertaining to a natural numbers $n$. Suppose that $P(0)$ is true, and suppose that whenever $P(n)$ is true, $P(n++)$ is also true. Then $P(n)$ is true for every natural number $n$.
to prove a proposition such as
$$1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}.$$
Here is an exercise about limitation
$$\lim_{n\to\infty}\left(\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\cdots+\frac{1}{\sqrt{n^2+n}}\right)=1\neq 0.$$
so that we couldn't use mathematical induction here. Notice that I don't mean you can exchange the order of the limits operation and addition operation.
An other example from Royden and Fitzpatrick's Real Analysis:
Proposition 5 The union of a finite collection of measurable sets is measurable.
Proposition 7 The union of a countable collection of measurable sets is measurable.
The authors use the definition to prove Proposition 7. However, why don't we just use the technology above with Proposition 5 to obtain Proposition 7?
