First, draw two line segments, each one unit in length, perpendicular to each other at an endpoint:
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The sum of their lengths is clearly two units. According to the Pythagorean theorem, the distance between the two endpoints equals the square root of the sum of each segment--in this case, the square root of two. Simple geometry, right?
But what if we divide each of those line segments in half, and shuffle them around a bit?
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The length of the entire path is still two units, four times one-half. No surprises there. Even if you do it again:
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...you'd still have a two-unit-long path. But if you continued this to infinity, you'd have an infinite number of infinitely-short line segments, which must be the same as a straight line between the two points, right? And this proves that the square root of 2 equals 2.
Well, no, it doesn't, for the simple but oft-ignored reason that infinity is not a number. But let's try this from a purely geometric approach, instead.
If you look at the third "zigzag" above, you can imagine two parallel straight lines that pass through all of the corners. As the line segments grow smaller and smaller, these parallel lines will draw closer and closer. But they will never actually touch each other. No matter how small you make those line segments, they will still have finite length. They will never form a single diagonal line because to do that, each line would need to be a single point--in other words, it has zero length. But you can't shrink it to zero length by chopping it in half repeatedly. No matter how many times you divide a number by two, it is still greater than zero.
The length of the "zigzag" will always be exactly two, even if you take the limit to infinity. It will never equal the square root of two, because you will never reach infinity.