# Why couldn't use the mathematical induction?

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?

• Let $a_1,a_2,\dots$ be a sequence. Sometimes one uses induction to prove that $a_n$ satisfies certain inequalities. These may be useful in finding $\lim_{n\to\infty}a_n$. – André Nicolas Aug 12 '13 at 4:21
• Another "induction limit proof": Proof of 1=0 by mathematical induction? – Martin Sleziak Aug 12 '13 at 8:18

Just because $P(n)$ is true for every natural number $n$ doesn't mean that $P(\infty)$ is true.
If $P$ is a proposition, then mathematical induction can show that $P(n)$ is true for every $n$ in the natural numbers. That is, it shows that a statement is true for every finite case - but a countable union doesn't have to be a finite union, so the usual form of induction won't give us a strong enough result. That is, we can prove the result for infinitely many finite cases, but no infinite ones.