Intereting Posts

Bijection between power sets of sets implies bijection between sets?
Finding sine of an angle in degrees without $\pi$
Lipschitz continuity implies differentiability almost everywhere.
Algorithm wanted: Enumerate all subsets of a set in order of increasing sums
Algorithm for real matrix given the complex eigenvalues
Is there a “continuous product”?
Why formulate continuity in terms of pre-images instead of image?
Philosophy or meaning of adjoint functors
Linear combinations of sequences of uniformly integrable functions
Does Chaitin's constant have infinitely many prime prefixes?
RSA in plain English
Subgroup of index 2 is Normal
Three Variables-Inequality with $a+b+c=abc$
To create a special matrix !!
Show uncountable set of real numbers has a point of accumulation

I’ve been struggling with the concept of proofs ever since I completed my introductory logic course “Axiomatic Systems”. In that course it seemed to be easy. We were pretty much just using various logical methods to prove the properties of real numbers. Now it doesn’t seem nearly as simple. I often find myself stumped when my real analysis text claims “by the principle of mathematical induction this is true,” offering no evidence whatsoever that the PMI implies anything. This makes me believe that I must not understand basic concepts like that very well. I really need a reference that properly explains these concepts to me, because even looking through my old axiomatic book I see no clear definition of induction. I’m not sure why I expected a good answer from that book. How could I when our assigned text was just over 70 pages for a full semester? (70 pages WITH examples in case you were wondering)

An explanation of mathematical induction would be appreciated, but I really would prefer book recommendations.

- Proof that there are infinitely many primes congruent to 3 modulo 4
- Show that $C(n,k) = C(n-1,k) + C(n-1,k-1)$
- Proof that if $\phi \in \mathbb{R}^X$ is continuous, then $\{ x \mid \phi(x) \geq \alpha \}$ is closed.
- Another way to go about proving Binet's Formula
- Singular value decomposition proof
- Linear surjective isometry then unitary

- What is the prerequisite knowledge for learning Galois theory?
- All real functions are continuous
- Learning general relativity
- Smooth spectral decomposition of a matrix
- Understanding differentials
- Distribution theory book
- Spectrum of $\mathbb{Z}$
- Book recommendation for ordinary differential equations
- A place to learn about math etymology?
- Prove if $f(a)<g(a)$ and $f(b)>g(b)$, then there exists $c$ such that $g(c)=f(c)$.

Induction is, in essence, a way of proving statements about integers. The “fact” that it works is really an axiom, and it says:

If $P(n)$ is some property of natural number $n$ which satisfies the following conditions:

- $P(1)$ is true.
- $P(n)\implies P(n+1)$ is true.

Then the statement $$\forall n\in\mathbb N: P(n)$$ is also true.

In practice, you prove statements by induction in two steps. For example, let’s have a look at how you can prove that $$2^0 + 2^1 + \dots + 2^n = 2^{n+1} – 1$$

In the first step, you prove that the statement is true if $n=1$. Therefore, you prove that $2^0 + 2^1 = 2^{1+1} -1$ which is trivial to prove.

In the second step, we **assume** that the statement is true for some $n$. Then, we try to prove that from this assumption, it follows that the statement is true for $n+1$. In our case, we **assume** that $$2^0 + 2^1 + \dots + 2^n = 2^{n+1} – 1$$ and we want to prove that

$$2^0 + 2^1 + \dots + 2^{n+1} = 2^{(n+1) + 1} – 1$$

This step is where mathematical creativity comes in. There is no blueprint to what you do next.

In our case, we prove it like this:

We know that $$2^0 + 2^1 + \dots + 2^n = 2^{n+1} – 1$$

But that means that $$2^0 + 2^1 + \dots + 2^{n+1} =\\ \left(2^0 + 2^1 +\dots + 2^n\right) + 2^{n+1} =\\ \left(2^{n+1} – 1\right) + 2^{n+1} =\\ 2\cdot 2^{n+1} -1 = 2^{n+1+1} – 1 = 2^{(n+1) +1} -1$$

which proves our equality.

If you want to really understand its full significance the principle of induction you should, in particular, not overlook that you can also show that if true for n also should apply to n-1 (not for “up” but for “down”).

This is a modality that goes unnoticed by many people.

I recommend the Godement’s book **Cours d’Algèbre** in its first six chapters and see in the fifth “Le raissonement par recurrence”.

- rational angles with sines expressible with radicals
- How do I show that $f: [0,1) \to S^1$, $f(t) = (\cos(2\pi t), \sin(2\pi t))$ is not a homeomorphism?
- Homomorphism between cyclic groups
- Showing $\log(2)$ and $\log(5)$
- How does e, or the exponential function, relate to rotation?
- Riemann integrability of continuous function defined on closed interval
- What are relative open sets?
- Solution to $e^{e^x}=x$ and other applications of iterated functions?
- How to see $\sin x + \cos x$
- Find the ratio of $\frac{\int_{0}^{1} \left(1-x^{50}\right)^{100} dx}{\int_{0}^{1} \left(1-x^{50}\right)^{101} dx}$
- Irreducible but not prime
- Elegant approach to coproducts of monoids and magmas – does everything work without units?
- Repairing solutions in ODE
- Is the set $P^{-1}(\{0\})$ a set of measure zero for any multivariate polynomial?
- Ideals in the ring of endomorphisms of a vector space of uncountably infinite dimension.