Component of Fibonacci Sequence

[leospyder]

Please excuse my total ignorance but can someon explain to me how the following part of a certain proof makes sense

We want to show that Fk+1 ≤ (7/4)^(k+1). Consider fk+1 = fk + fk−1 (We can do this
as k +1 is at least 2; see the comment following the basis) < (7/4)^k +(7 /4)^(k−1) (by the Induction Hypothesis;
notice how the stronger hypothesis comes in handy here.)

The parts I bolded in red are mainly the things I don't understand. I plugged in the (7/4)...part into my calculator and did not get the alleged answer I was supposed to get if it were simply (7/4)^k+1. Can someone please enlighten me?

 

[matt grime]

If a<b, and c>0, then ac<bc (and no that is not cryptic - you end up with something that you wish to show is less than something else - there is no reason to suppose they are equal, nor is it necessary. If I want to show something is less than 4 and I show it is less than 3 I've shown it is less than 4, for example).