Charles Lyell

Uniform

For the statement "convergence implies boundedness," the converse statement would be "boundedness implies convergence."

So, we are asking if "boundedness implies convergence" is a true statement.

Pf//

By way of contradiction, "boundedness implies convergence" is false.

Let the sequence (Xn) be defined as
Xn = 1 if n is even and
Xn = 0 if n is odd.
So, (Xn) = {X1,X2,X3,X4,X5,X6...} = {0,1,0,1,0,1,...}

Note that this is a divergent sequence.
Also note that for all n, -1 < Xn < 2

Therefore, the sequence (Xn) is bounded above by 2 and below by -1.

As we can see, we have a bounded function that is divergent. Therefore, by way of contradiction, we have proven the converse false.

Q.E.D.

The Principle of Uniform Processes states that the same processes acting on the earth today have acted on the earth throughout its history. It states that laws of physics and chemistry have not changed through time. Therefore, the processes that work today on the earth has always been acting, although not necessarily at the same rates that they do now.