Question

prove that the bounded sequence in a closed interval is not continuous in that interval​

Answer 1

Answer:

The boundedness theorem

This result explains why closed bounded intervals have nicer properties than other ones.

Theorem

A continuous function on a closed bounded interval is bounded and attains its bounds.

Proof

Suppose f is defined and continuous at every point of the interval [a, b]. Then if f were not bounded above, we could find a point x1 with f (x1) > 1, a point x2 with f (x2) > 2, ...

Now look at the sequence (xn). By the Bolzano-Weierstrass theorem, it has a subsequence (xij) which converges to a point α ∈ [a, b]. By our construction the sequence (f (xij)) is unbounded, but by the continuity of f, this sequence should converge to f (α) and we have a contradiction.