Thread: Induction on sequences

The sequence of positive numbers is such that
and


By considering , prove by induction that for all

My attempt N.B To make this easier to read I'm skipping all the official statements of the induction process


Let

It is given that and positive so

Hence is less than

Assume true for


Test at

End of attempt

To finish this step confuses me. I implore the forum for assistance.

Originally Posted by I-Think

The sequence of positive numbers is such that
and


By considering , prove by induction that for all

My attempt N.B To make this easier to read I'm skipping all the official statements of the induction process


Let

It is given that and positive so

Hence is less than

Assume true for


Test at

End of attempt

To finish this step confuses me. I implore the forum for assistance.

You're so close!

You have a nice way of writing 4-u_{k+1} in terms of u_k. So now you need to show that , which in turn shows that 4-u_{k+1}<4. This should be easy for you, since you have assumed that u_k < 4. Use that to show that the above fraction is < 4, thus 4-u_{k+1}<4, thus u_{k+1}<4. This shows that u_{k} < 4 implies u_{k+1}<4, which is the proof.