Question: What is a better way of saying "in order to demonstrate this, I will..."? I already use "to that effect" in the same paragraph. I'm looking for something as compact and concise as "to that effect" that will also be a good transition for the beginning of a new paragraph.

Answer: 
Is the sentence in question immediately followed by the character actually performing the demonstration? In that case: 

"Observe."
(Does thing). 

However, the whole thought "in order to demonstrate this, I will..." might be completely unnecessary. If the character already talked about the thing, just having them do the thing should save words.
"I'll show this (is true, is so, isn't so) by. . .