About Math notation: the set of the first $n$ natural numbers

notation

You can define $[n]$ however you want, so it can be true. Despite that, it is a common notation for the set $\{k\in \mathbb{N} : k\leq n\}$, yes.

This notation is used more often on Elementary Set Theory and Discrete Mathematics. Unfortunately analysts don't use it much. I've never seen it being used in Abstract Algebra or Linear Algebra either.

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