Which of the following statements are true?
- Every index can uniquely be matched to a sequence element .
- Every sequence element can uniquely be matched to an index .
- A sequence being neither bounded from above nor from below does not exist.
- A sequence increasing and decreasing monotonically does not exist.
- A sequence increasing and decreasing strictly monotonically does not exist.
- A sequence increasing monotonically and being bounded from above does not exist.
- False. A constant sequence is a great counterexample. As all sequence elements , , have the same value, you can match this value to every index and not only one unique index.
- False. The sequence is neither bounded from above nor from below.
- False. Every constant sequence is monotonically increasing and decreasing. Only „strict monotonicity“ implies that the sequence elements need to be unequal.
- False. The sequence with for all is bounded from above, as holds for all . Furthermore, it's monotonically increasing. For all , we have and therefore .
Exercise: Find a sequence[Bearbeiten]
Find a sequence fulfilling the following conditions:
- for all
- for all odd
Write down an explicit and a recursive formula for your sequence!
One possible sequence is
An explicit formula of this sequence is given by
A recursive formula of this sequence is given by and for all :