Interval

We use the idea of an interval in the technical definition of an interval.

As luck would have it, building this in for one reason can help us out with a second reason.

Let’s say we can change length continuously from a starting value to stopping value.

Assume we start at 1.4 and 1.6 is our stopping value. At some point between start and stop the length was 1.5 because 1.5 is on the interval [1.4, 1.6].

Now assume we want the length \sqrt 2 which is an irrational number. We don't know the exact value when we write it out, but we know at some point we hit it on our interval of [1.4, 1.6] because the approximation with four digits is 1.4142.

Now, the definition of a limit as envisioned by Isaac Newton said we had to have the ability to approach as close as we wanted to the suspected function by changing the value of the independent variable on an interval approaching some "value of interest".

What if we are using a method where we can't hit the requested value? What if the best we can do is say that our first attempt wasn't close enough, and our second guess was closer than the request.

Given what we have above, we can argue that as we changed on the interval, at some unknown point we equaled the desired number.

Now that we know this, we can say that if we are approaching on an interval, our knowledge claim that we are closer gives us that at some point we had it.

The reason we ask for this, we might have a situation where we are approach 3 with attempts as follows:

• 2.9
• 2.99
• 2.999
• 2.9999
• and so on, as many 9's as we want

If someone asks for 2.95, we can argue we got it when we went from 2.9 to 2.99.

It is typical on a homework problem for you to discover that the strategy you are using for making closer guesses will make your calculation do something like 2.99, 2.999, 2.9999 and so on, so that you see you can generate as many 9's as you want.

What we are telling you here is that you have proven at the end of the interval the calculation will be 3.