Can some one explain this theorem?

Theorem Every limit point of the range set of a sequence is limit point of a sequence. Solution Let S be range set of sequence {a,) i.e. S = range of fa,} Let a e S for &> 0 (a E, a + E) NS{a} has infinite no. of points Let q e (a - &, a + e) N S{a} g e (a-E, a + E) and qe S As eS g a for some k e N So a, e (a E, a + e) Hence, (a E, a + 8) Contains infinite no. of terms of sequence Proved a is limit point of sequence {a} Remark : Areal no. is a limit point of sequenceit appears in the sequence infinite many times or it is the limit point of the range set.