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In an idealized situation you are able to focus closer using a pinhole
because only one light ray from any given point on the object hits the
lens of your eye. Since there is only one light ray there is no problem
with focusing - the lens acts to redirect the rays not focus them and
there is an in-focus image at all distances from the lens. This is why
so many diagrams showing the eye and a pinhole negate the effects of
the eye's lens when doing ray tracing (which really mucks up students,
BTW).
With a pinhole, magnification the image size is proportional to the
object distance, such that if the pinhole is half way between your eye
and the object, there is zero magnification, and if your eye is twice
as far away the object will appear twice as large. I have to say that I
have never been able to actually see this, but this is what the math
claims, and I blame my inability to see through a small enough pinhole.