Hi iwish123,

Show by induction that

If

then

hence

As

then

So, being < 5causesall subsequent terms of the sequence to be < 5 also.

A non-inductive proof is as follows...

The index is a geometric series, first term=1, common ratio = 0.5 (for the part in brackets)

so the index of 5 is 1.

Hence cannot reach 5 as a finite number of terms will not reach the sum to infinity as all the terms are positive.