Intereting Posts

Solve $(x+1)^n=(x-1)^n$, assuming $x$ is a complex number and $n>0$.
If a matrix is triangular, is there a quicker way to tell if it is can be diagonalized?
What is $k_{\text{max}}$?
How to find inverse of 2 modulo 7 by inspection?
Proving the intersection of distinct eigenspaces is trivial
Partial sums of $\sin(x)$
Where is the fallacy? $i=1$?
Prove that $x=0.1234567891011\cdots$ is irrational
A double sum with combinatorial factors
A method of finding the eigenvector that I don't fully understand
Proving irreducibility of $x^6-72$
zeroes of holomorphic function
Finiteness of the dimension of a normed space and compactness
If $p\mid 2^n-1$ , then how to prove $l(n) \lt p$
$k$-tuple conjecture.

When the goal is $∀n\in\Bbb N ∀m\in\Bbb N (n \ge m \rightarrow H_n-H_m \ge {n-m \over n})$, I can begin the proof with “Let n and m be arbitrary. Suppose n and m are natural numbers” or “Let n and m be arbitrary natural numbers.”

The boundary between “let” and “suppose” feels blurry. When do I use “let” and “suppose” in a math proof?

- question on translation of operator proof
- Proof problem: Show that $n^2-1$ is divisible by $8$, if $n$ is an odd positive integer.
- Combinatorial Proof -$\ n \choose r $ = $\frac nr$$\ n-1 \choose r-1$
- For which natural numbers $n$ is $\sqrt n$ irrational? How would you prove your answer?
- Prove the inequality $n! \geq 2^n$ by induction
- Proof related to Harmonic Progression

- A subset of a compact set is compact?
- The number of vertices in a polytope is finite
- How to relate areas of circle, square, rectangle and triangle if they have same perimeter??
- Given a rational number $x$ and $x^2 < 2$, is there a general way to find another rational number $y$ that such that $x^2<y^2<2$?
- End of step symbol
- How does a non-mathematician go about publishing a proof in a way that ensures it to be up to the mathematical community's standards?
- Is there a geometrical method to prove $x<\frac{\sin x +\tan x}{2}$?
- Prove $\bigcap \{A,B,C\} = (A \cap B) \cap C$
- Showing $a^2 < b^2$, if $0 < a < b$
- Rigorous Proof?: Proving Cauchy Criterion of Integrals

“Let $n$ and $m$ be arbitrary natural numbers” assigns a meaning to $n$ and $m$ whereas “Suppose $n$ and $m$ are natural numbers” makes an assumption on the meaning of $n$ and $m$. In terms of proof writing, the difference between the two is fairly arbitrary since if one makes a supposition on an as of yet defined variable it’s assumed to be definitive. The biggest distinction between the two is that “suppose” does not necessarily assume the concept exists, as in the case of proof by contradiction.

As commented by user2520938, I use “let” when considering an object which I suspect exists, and “suppose” when considering an object I think does not exist (so as to derive a contradiction).

- “Suppose $\sqrt{2} = \frac{p}{q}$ for $p, q$ integer.”
- “Let $n \in \mathbb{Q}$ be written as $\frac{p}{q}$ in its lowest terms.”
- “Let $G$ be a simple group.”
- “Suppose $G$ were a simple group of order 50.”

I think you use ‘let’ as in

Let $x$ be s.t. $P(x)$

When you know for sure that this is possible. On the other hand you use

Suppose $x$ is s.t. $P(x)$

is used when you’re not yet sure/are going to derive a contradiction from here.

For example, when you want to proof that

$x$ is limit point of $A\subset \mathbb{R}$ iff $\exists\{x_n\}\subset A-\{x\}$ with $x_n\to x$

Then for the $\Leftarrow$ direction I might start by saying something like

Suppose $x$ is s.t. there is a sequence $\{x_n\}\subset A-\{x\}$ with $x_n\to x$ but $x$ is not a limit point of $A$

And derive a contradiction. I might also start by saying

Let $x$ be s.t. there is a sequence $\{x_n\}\subset A-\{x\}$ with $x_n\to x$

And then just prove $x$ is a limit point directly.

Again, I’m not sure this is the way everyone thinks about it, but to me it makes the most sense. It also makes it clear to the reader what direction the proof is going in.

- equation of a curve given 3 points and additional (constant) requirements
- Complement of $c_{0}$ in $\ell^{\infty}$
- Prove that the limit $\displaystyle\lim_{x\to \infty} \dfrac{\log(x)}{x} = 0$
- Is it possible for a function to be in $L^p$ for only one $p$?
- Efficient low rank matrix-vector multiplication
- A question about proving that there is no greatest common divisor
- ODE Laplace Transforms: what impulse brings an oscillating system to rest?
- A question about differentiation
- What is the expected number of trials until x successes?
- If $a_1,a_2,\ldots, a_n$ are distinct primes, and $a_1=2$, and $n>1$, then $a_1a_2\cdots a_n+1$ is of the form $4k+3$.
- Evaluating an integral across contours: $\int_C\text{Re}\;z\,dz\,\text{ from }-4\text{ to } 4$
- Universe cardinals and models for ZFC
- Do runs of every length occur in this string?
- Asymptotics of $\sum_{n\leq x}d(2n)$
- To whom do we owe this construction of angles and trigonometry?