More points in a line segment of length 1 than one of length sqrt(2)

Consider the line segment A connecting (0,0) and (1,1) and the line
segment B connecting (0,0) and (1,0). A has length sqrt(2), B has
length 1.

The points of A can be put into one-to-one correspondence with a subset
of the points of B as follows: every point (x,x) on A maps to a point
(xx, 0) on B. For example, (.526, .526) on segment A maps to (.526526,
0) on B. (0,0) on A is arbitrarily mapped to (0,0) on B, and (1,1) on A
is mapped to (1,0) on B. Every point on A is therefore mapped to a
point on B.

However, there are points on B that cannot be mapped to points on A,
such as (.534, 0) and (.335, 0).

Since all the points on A can be put into one-to-one correspondence
with points on B, and there are more points on B that are not in this
correspondence, it follows that B has more points than A.

What is wrong with my reasoning?

.

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