Prove that a is a limit point of a set D ⊂ R^p iff there is a sequence of points in D (not equal to a) which converges to a.
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This can be taken as the definition of a limit point. I'll have to assume that what you are taking as a definition is the following:
a is a limit point of D provided every deleted neighborhood S of a intersects D.
I'll also assume the standard topology on IR^p.
Suppose that a is a limit point of D. Then for every integer n the deleted ball B_0(a, 1/n) intersects D. Hence we can choose a point x_n in B_0(a, 1/n) ∩ D for each n. As the radius of the balls is vanishing, it is clear that the sequence {x_n}, which is contained in D, converges to a.
Conversely, suppose there exists a sequence {x_n} ⊂ D such that x_n -> a and x_n ≠ a for each n. Let S be any open neighborhood of a. Since S is open, we can find ε > 0 sufficiently small such that the ε-ball B(a, ε) = {x | ||x - a|| < ε} is contained entirely within S. As x_n->a, we can find N sufficiently large such that for all n ≥ N, x_n ∈ B(a, ε) ⊂ S. Hence S - {a} ∩ D ≠ Φ. As S was arbitrary, we can conclude that a is a limit point of D.
Notational Note: I hope this was clear:
B_0(a, 1/n) = {x | 0 < ||x - a|| < 1/n}
a is a limit point of D provided every deleted neighborhood S of a intersects D.
I'll also assume the standard topology on IR^p.
Suppose that a is a limit point of D. Then for every integer n the deleted ball B_0(a, 1/n) intersects D. Hence we can choose a point x_n in B_0(a, 1/n) ∩ D for each n. As the radius of the balls is vanishing, it is clear that the sequence {x_n}, which is contained in D, converges to a.
Conversely, suppose there exists a sequence {x_n} ⊂ D such that x_n -> a and x_n ≠ a for each n. Let S be any open neighborhood of a. Since S is open, we can find ε > 0 sufficiently small such that the ε-ball B(a, ε) = {x | ||x - a|| < ε} is contained entirely within S. As x_n->a, we can find N sufficiently large such that for all n ≥ N, x_n ∈ B(a, ε) ⊂ S. Hence S - {a} ∩ D ≠ Φ. As S was arbitrary, we can conclude that a is a limit point of D.
Notational Note: I hope this was clear:
B_0(a, 1/n) = {x | 0 < ||x - a|| < 1/n}