# Question on geometrical proof of Geometric Series

In the drawing on Geometric Series Proofs: An Annotated Bibliography, using Fig. 4 on the page, Please explain why it is said that "ON is the limit of the sum 1+x+..."? Thank you.

Edit, I guess what through me off was the word "limit"!

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Notice that the horizontal lines of length $1,x,x^2,x^3,\ldots$ union to form a line of length $ON$.
There are two ways to travel the horizontal distance from $O$ to $N$. The easiest way is to go via the line $ON$. Alternatively, you could follow the staircase, only counting the movement in the horizontal direction. The first stair has horizontal movement $1$, the second has horizontal movement $x$, and so on. However, once you've arrived at $N$, it shouldn't matter which way you went, the distances are the same. Therefore (the length of the line segment) $ON$ is the same as the limit of the sum $1 + x + x^2 + \dots$