Question

Imagine you had a bar of gold and decided to cut in half. You repeated this process until eventually you could no longer cut the gold in half without splitting a single atom of gold. This would prove that gold is a(n)
a. mixture.
b. element.
c. compound.
d. mixture or compound.

Answer 1

So we observed that no matter how small we make it by cutting, we get only gold and the smallest particle is also gold. So gold is an element

Answer 2

This would prove that gold is an element.  No matter how far down you
examine it, you never find any particles of anything except gold.

An example of a different case is salt. 
-- Imagine you had a block of salt and decided to cut it in half.
-- If you repeated this process, then eventually, at some point, you'd have
a tiny particle of salt in front of you, just like before.  BUT ...
-- Just as you were getting ready to cut this one in half, you'd notice that this
particle of salt is different.  It's one atom of sodium stuck to one atom of chlorine,
and if you cut it in half, you would not have ANY salt.   <span />

This would prove that salt is a compound, made of atoms of two or more elements.