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.

Thus, .

This completes the proof.

You can try doing the other problems now.