3333-Solns-Assignment6 - MATH 3333 Homework Assignment#6...

MATH 3333
Homework Assignment #6
Due date: Friday, October 12
NAME (Print):
1. ALWAYS TRUE, SOMETIMES TRUE, NEVER TRUE. If your answer
is “Always True,” justify by quoting a definition or theorem, or by giving a proof.
If your answer is “Sometimes True,” give an example where it is true and an
example where it is false. If your answer is ”NEVER TRUE,” justify by quoting
a definition or theorem to the contrary, or by giving a proof.
(a) If ( s
n
) is a Cauchy sequence, then ( s
n
) is monotone.
(b) If ( s
n
) is a convergent sequence and ( t
n
) is bounded, then ( s
n
· t
n
) is
a convergent sequence.
1

(c) If ( s
n
) and ( t
n
) are sequences such that s
n
> t
n
> 1 for all n , and
( s
n
· t
n
) converges, then both ( s
n
) and ( t
n
) converge.