Thinking about infinity

[robvalue]

Just as an aside:

When we use the concept of infinity in mathematics, it is usually employed as a limit. We allow a number to tend towards infinity, and we see what effect that has.

For example, 1/n tends towards zero as n tends towards infinity.

The technical way of writing this is:

For any positive number k, we can find a number m so that 1/n < k for all n >= m (greater than or equal to)

[In general we would use |1/n| which means to ignore any negative values, but we know in this case, dealing with positive values of n, 1/n is always positive anyway.]

In other words, however small we want the value of 1/n to be, we can find a point so that the values of 1/n that follow are all below that number. It gets arbitrarily close to zero, and so we say that it becomes zero as the limit is taken of n=> infinity.

It's important that all values of 1/n are below this value of k after this point we find, or else we might be dealing with an oscillating function that jumps in and out of the required zone.

This value of m can be easily found:

We want 1/n < k

Multiply by n to give 1 < nk

Divide by k to give 1/k < n

Turning that around gives n > 1/k

So the value of m will be the first whole number greater than 1/k. Let's say k = 0.000001 (a millionth). Then we need

n > 1/0.000001

n > 1,000,000

So 1/n < 0.000001 for all n >= 1,000,001

Did that make any sense to those not familiar with the subject?