Show that convergent sequence is bounded?

Answer 1

Wrong answer above. A limit is not the same thing as a limit point. A limit of a sequence is a limit point but not vice versa. Every bounded sequence does have at least one limit point. This is one of the versions of the Bolzano-Weierstrass theorem for sequences. The sequence {(-1)^n} actually has two limit points, -1 and 1, but no limit.

Answer 2

A sequence, {xn}, is said to converge to a limit x if, given any e > 0, however small, there exists an integer k, such that abs(xk - x) < e for all n >=k. That is all value of the sequence, from xk onwrads, are no further from x than e.abs(xk - x) < e => -e < xk - x and x - xk < e which implies that xk is bounded.

Answer 3

It cannot be proven because it is not true.

The sequence t(n) = (-1)n is bounded: by -1 and 1 but it does not have a limit point: it oscillates.

Answer 4

The best answer is no answer