\documentclass[border=9,tikz]{standalone}
\begin{document}
\tikz{
\clip(0,0)rectangle(10,10);
\foreach\x in{0,...,10}{
\foreach\y in{0,...,20}{
\draw(-\x0,-\y0)--+(100,141.421);
}
}
}
\end{document}

A proof that this works
\tikz{
\clip(0,0)rectangle(10,10);
\foreach\x in{0,...,3}{
\foreach\y in{0,...,10}{
\draw(-\x0,-\y0)--+(100,141.421);
}
}
\foreach\x in{1,...,5}{
\draw[blue,dotted]({mod(7.07106*\x,10)},0)--+(0,10);
}
\foreach\x in{1,...,3}{
\draw[red,dotted](0,{mod(4.14213*\x,10)})--+(10,0);
}
}

An animation that shows how line grow
\foreach\frame in{0,...,40}{
\tikz{
\clip(0,0)rectangle(10,10);
\foreach\x in{-5,...,5}{
\foreach\y in{-8,...,8}{
\fill[shift={(\x0,\y0)},scale=\frame/10]
(-.2,.2)--(10,14.1421)--
(.2,-.2)--(-10,-14.1421)--cycle;
}
}
}
}

Bonus

\clip(0,0)rectangle(1,1);
and then draw slope-√2 lines that passes integral coordinates.