Is it possible to go over all lines of an infinite grid with a pencil without lifting it or going over a drawn line ?
The pencil can cross through a segment already drawn but cannot go over an already drawn line.
After doodling around, I have the feeling it is not possible. If indeed it is not possible what demonstration exist of this result ? If it is possible then can you show a way of drawing the grid ?
Easily one sees that drawing a sub-grid of $m\times n$ is possible for all values of m and n, by drawing vertical lines and then horizontal lines, without removing the pencil.
Here is an example of a sub-grid of $3\times 6$.
Yes, it is possible, using the method illustrated in this image.
Referring to Especially Lime's comment, this answer used to have an alternate proof which Lime noticed was fallacious, so I deleted that proof.