Complete Metric Space | Lecture 7 | Result

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If p is a limit point of set A then there is a sequence of distinct points of A converging to point p.
  1. Ranjan Khatu
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How can we be sure that x_1 is not equal to x_2? It is possible that x_1 is less than 1/2 units away from p and is the only point that is in the two balls, then certainly the intersection is non-empty but we don’t get two distinct points.

Should we not remove that possibility completely by considering the new ball as entered p with radius d(x_1, p) / 2? That way we know the next point x_2 cannot be x_1?

shaheerziya