A non-closed set M does not contain all its limit points, hence a sequence within M can be constructed to converge to a limit point outside M, such as 1/n converging to 0 in the open interval (0,1).
To show that if a set M is not closed then there is a sequence of points of M that converges to a point p that is not in M, we must understand the definition of a closed set. A set is closed if it contains all its limit points. Therefore, if a set is not closed, there exists at least one limit point that the set does not contain. We can construct a sequence within M that converges to this limit point; this sequence will approach the point p, but since M does not contain p, by definition, the sequence will be converging to a point outside of M.
For example, consider the open interval M = (0,1). This set is not closed because it does not include its limit points 0 and 1. We can take a sequence like 1/n, which is in M for all n > 1, and observe that as n grows larger, 1/n gets closer to 0, which is not in M.
So, a set that is not closed can be demonstrated to have a sequence within it that converges to a point outside of the set, confirming the definition of a non-closed set.