Show by induction that
So, being < 5 causes all 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.