TITLE: What does it mean by a density argument?
QUESTION [7 upvotes]: When I read some books, it states "a density argument", what does it mean? When applying to symmetric function, what does it mean?

REPLY [6 votes]: For the sake of an answer (thanks to the comment above):

It is difficult to answer without a precise context, but usually "by a density argument" means that you had proved a statement for all elements $x$ in a dense subset $A$ of a topological set $X$, and you extend the statement to the whole $X$ using density. For example, if you prove that a continuous function $f$ is constant on a dense subset $A$ of $X$, then $f$ is constant on $X$ by density.