Bolzano-weierstrass theorem theorem: every bounded sequence has a limit point. proof: let (a.) be a bounded sequence. let s = {a,: n e n} be its range. since t

BOLZANO-WEIERSTRASS THEOREM Theorem: Every bounded sequence has a limit point. Proof: Let (a.) be a bounded sequence. Let S = {a,: n e N} be its range. Since the sequence is bounded, therefore, its range S is also bounded. Case I. Let S be a finite set. Then there must exist at least one element a e S such that a = a for an infinite number of values of n. For any 8> 0, the nbd. (a - 8, a + e) of a, contains a, o, for an infinite numb of n. Therefore, a is a limit point of (a,) Case II. Let S be an infinite set. The range S being an infinite bounded set has a limit point, say p, So each nbd( of p contains an infinite number of elements of S i.e. a, e (p E, p + 8) for an infinite number of values of n. Hence p is a limit point of (a, Remark An unbounded sequence may or may not have a limit point. Counter example, Since a, = n is an unbounded sequence with no limit point and even; a=n, if n is odd is an unbounded sequence with a limit point 1.