Is the Cantor set made of interval endpoints?

Solution 1:

Because the Cantor set includes numbers which are not the endpoints of any intervals removed. For example, the number $\frac{1}{4}$ (0.02020202020... in ternary) belongs to the Cantor set, but is not an endpoint of any interval removed.

Solution 2:

You cannot do that because a countable set can have an uncountably many limit points. The points in the Cantor set are limit points of these endpoints.

For example, the real numbers are all limit points of the rational numbers. If between every two real numbers there is a rational number, but we still can't establish that the real numbers are countable.