Show, rigorously, that the geometric series $a + ar +ar^2 + \cdots + ar^{n-1} + \cdots$ converges if and only if |r| < 1.
Also, show that if |r| < 1, the sum is given by $S = \frac{a}{1-r}$.
Note(s):
Question 1, First part: if |r| < 1, show (rigorously) that the geometric series above converges.
Update: I'm reposting this question as the accepted answer on the possible duplicate question doesn't provide a rigorous [$\epsilon$-$\delta$] derivation of an important step, namely, $\lim_{N\to\infty}r^{N+1} = 0$ if $|r|\lt 1$. [Translated to the current question's context, $\lim_{n\to\infty}ar^n = 0$ if $|r|\lt 1$].
Update 2: The problem of the above issue has been addressed.
Question 1, Second part: if the geometric series above converges, then show (rigorously) that |r| < 1.
-- Any hints would be appreciated. --