Let us prove: .
(If are bounded then so is ).
Let , well this means .
The sequence, is non-increasing therefore the limit, ,
is the greatest lower bound, so for all .
For we see therefore is a lower bound for .
This means, for all .
Likewise if then for all .
We are told that which means .
Thus, we can write inequality, for all .
This means that, for all .
Thus, is a lower bound for the sequence .
But this 'inferior sequence' is non-increasing so it has a limit.
This completes the proof.
You can try doing the other problems now.