I'm trying to prove by way of induction that:

The base case is obviously true as

I then took the inductive step and came up with the following:

I then figured that if (assumption), then if the amount that the LFH is being multiplied by each time is > the amount that the RHS is being multiplied by each time, then the LHS must always be > RHS and the statement is true.

So if:

then the original statement is true.

Since then the LHS multiple is always > 4 which is > 2 (RHS multiple), therefore the LHS is always greater than the RHS and the statement is true.

Is this a valid proof? What are some other ways I could prove this?

Thanks.