The Convergence of Sequences Section 2.2 Monotone Sequences, the Bolzano-Weierstrass Theorem, and the Nested Interval Theorem. If a sequence does converge to a , that limit is unique. Suppose the sequence converged to both a and b , where a is not equal to b . Thus our original assumption is false, so a must be equal to b . Read proof on your own.
Every convergent sequence is bounded.
Says that a nonnegative sequence can only converge to a nonnegative limit. Then the same argument applies to the first one. Theorem 2.9 Squeezing Principle. Section 2.2 Monotone Sequences, the Bolzano-Weierstrass Theorem, and the Nested Interval Theorem. A monotone sequence converges iff it is bounded.