A question about Infinity. Does it exist?

Let's look at the sequence of natural numbers which I think is the "simplest" infinity we can talk about.

Natural numbers: 1, 2, 3,...

Observe how successive numbers "increase"

a) 1 to 2 the quantity has doubled (2 = 2 × 1)
b) 2 to 3 : (3 = 1.5 × 2)
c) 3 to 4 : (4 = 1.33... × 3)
d) 4 to 5 : (5 = 1.25 × 4)
.
.
.as you can see the factor (numbers in bold) is decreasing and approaching a limit which is 1. Look at larger numbers below:

e) 9999 to 10000 : (10000 = 1.001... × 9999)
f) 99999 to 100000 : (100000 = 1.0001... × 99999)

The pattern suggests that eventually there will be two very very large numbers A and B such that:

1. B = A + 1 (B is the next number we getting by adding 1 to A)
2. B = A × 1 = A (the pattern I showed you suggests that 1 is the limit of the factor by which a number increases in bold)

In other words there is a largest number and infinity doesn't exist.