Hope this helps

grad = Grad[fun, {x, y}]
(*gradFunc gives the gradient at a coordinate*)
gradFunc = grad /. Thread[{x, y} -> N[{##}]] &;
(*stepfunc takes two arguemnts #1=coordinate; #2=stepsize*)
(*coor + step * grad[coor]*)
stepFunc = (#1 + #2 gradFunc @@ #1) &;

(*gradFunc takes the coordinate*)
gradFunc[x1, y1]
(*stepfunc takes the coordiante and the stepsize and calculates the \
new coordinate*)
{x2, y2} = stepFunc[{x1, y1}, s]

The NestList is just the easy way to repeatedly apply a function to a previous output to generate the list of coordinates.

These are the same functions without using short notations or pure functions.

gradFunc2[xi_, yi_] := Replace[grad, {x -> xi, y -> yi}]
stepFunc2[coor_, step_] := coor + step Apply[gradFunc, coor];
(*OR*)
stepFunc3[coor_, step_] := coor + step gradFunc[coor[[1]], coor[[2]]];

Your solution is absolutely perfect and does exactly what I wanted. Now, I really need to understand exactly what your code is doing and not being as knowledgeable in pure functions as you apparently are, it is causing me a few issues in understanding your solution, so please bear with me.

I see that you are passing 2 parameters into stepFunction from the function - points. In the first iteration, it seems that you are sending two parameters: (the starting point (3,4) and the stepSize of 2) to the function - gradFunc. What I don't understand is what are you doing with the two parameters in the stepFunction before you execute the gradFunc.

Thanks Again,

Mitch Sandlin