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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 $3\times 6$. enter image description here

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    $\begingroup$ If there is a "starting point", then it is impossible. You will have to end at the same point you started if you want to cover all four directions from that point. But covering the whole plane is an infinite process, so it will never end. Therefore you cannot end at the starting point. $\endgroup$
    – Polygon
    Commented Mar 5 at 18:59

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Yes, it is possible, using the method illustrated in this image.

enter image description here

Referring to Especially Lime's comment, this answer used to have an alternate proof which Lime noticed was fallacious, so I deleted that proof.

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