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Number of elements in the set {1,…,n}*{1,..,n}

Writing $[n]$ for the set $\lbrace1,2,...,n\rbrace$, let $P_n$ denote the product set $[n].[n]$, i.e. $$ P_n = \lbrace ab : a,b \in [n]\rbrace .$$

Since the set $[n]$ is quite far from looking like a geometric progression, one would suspect that the set $P_n$ is quite large. Let $$ c_n = \frac{|P_n|}{n^2} .$$

I was hoping to find out what the asymptotics are for $c_n$; I suspect the answer is well known to additive combinatorialists. In particular, is $c_n$ bounded away from $0$ or is it $o(1)$?