[Tex/LaTex] File ended while scanning use of \align

errors

I have seen this question posted before and the answer was to eliminate blank space before \end. However I did this and I am still getting an error.
Here is my code for this portion:

Lemma 1: Let $f:  \mathbb{Z}_{>0} \rightarrow  \mathbb{Z}_{>0}$. 
Let $n \in \mathbb{Z}_{>0}$. Let $\sum\limits_{d|n} f(d)$ be the sum over all divisors of $n$. Then $F(n)=\sum\limits_{d|n} f(d)$ is also a multiplicative function.  
\bigskip
$Proof.$ Let $F(n)=\sum\limits_{d|n} f(d).$ Let $m,n \in \mathbb{Z}_{>0}, gcd(m,n)=1$. Additionally, let $d=rs$, where $r$ and $s$ are divisors of $m$ and $n$ respectively.

\begin{align}
F(mn)=\sum\limits_{d|mn} f(d) \\
F(mn)=\sum\limits_{r|m,s|n} f(rs) \\
F(mn)=\sum\limits_{r|m,s|n} f(r)f(s) \\
F(m)F(n)=(\sum\limits_{r|m)f(r))(\sum\limits_{s|n)f(s))
\end{align}

Best Answer

You have no closing }'s (2 times) in the line:

F(m)F(n)=(\sum\limits_{r|m)f(r))(\sum\limits_{s|n)f(s))

Additionally, it is a bad manner to write:

$Proof.$ Let $F(n)=\sum\limits_{d|n} f(d).$ Let $m,n \in \mathbb{Z}_{>0}, gcd(m,n)=1$. 

It would be better to use (amsthm) \begin{proof} ... \end{proof}. Additionally, we have macro \gcd and we should write $., not .$, even if the difference is invisible, because it shows a structure: some words, which may be math formulas, and a fullstop, which ends the sentence.

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