MATH1071 - Sequences and their limits

The limit of a sequence

For all epsilon > 0, there exists some N contained in N such that if n>N then [an-L]< epsilon

Bounded monotone theorem

A monotone sequence converges if and only if it is bounded and will converge if and only if the supremum = infimum.

• Must show it is monotonic

• Bounded by something

Proof: Squeeze Theorem

Triangular inequality for bn - L, use telescoping, drop the absolute value. sub in cn, telescoping and put in epsilons

Proof: product law

Triangualar inequality, telescoping, bounded theorem, fix epsilon, choose the max N between N1 and N2, sub in and cancel.

Proof: Bounded-monotone theorem

Suppose a sequence is monotone increasing, define something as the supremum of sequence which occurs because the sequence is boinded, apply definition of limit of supremum sequence.