How do I know when to use “let” and “suppose” in a proof?

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?

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“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.