The question was inspired by this excellent blog post re: the equation 0.999... = 1.
I see a connection between:
· Zeno’s Dichotomy (1/2 + ¼ + 1/8 + … = 1)
· 0.999… = 1
· The derivative as a limit (i.e., ever approaching a value, never reaching it, after all “at” a point you get 0/0)
Enter the epsilon-delta definition of a limit:
· But this doesn’t resolve any of the above “paradoxes”
· Rather, it specifies what we mean by “limit” and “convergence”
· Absent is an explanation for why we are justified in equating the sum of an infinite series with its limit. For example, you often read that calculus “solves” Zeno’s Dichotomy paradox by defining it away: the truth that 1/2 + ¼ + 1/8 + … = 1 follows from our definition of limit
The blog post made me wonder: does the Archimedean property help provide us with the missing explanation?
· If we have found the limit of the function in question, that means we have won the “epsilon-delta” game (for every epsilon, we can find the appropriate delta and so forth)
· Then the difference between the limit and the function can be shown to be less than any given epsilon > 0
· Borrowing a line from the blog post, another way to say that is the difference between the function and the limit is some positive number that is less than or equal to 1/10^k for every k
· But by Archimedes property, this is impossible – no such number can exist
· The function and the limit must therefore be the same number
What’s your reaction to the above line of reasoning? To me, it seems to provide the reason why we’re able to equate an infinite series with the limit to which it converges.