lim(n→∞) r^n = 0 (if |r| < 1)

lim(n→∞) r^n = ∞ (if |r| > 1)

using the ε-δ technique

when I'm focusing on 0 < |r| < 1 something weird happens

|r^n - 0| < ε ⇒ n > N

|r^n| < ε ⇒ n > N

n ln |r| < ln ε ⇒ n > N

n > (ln ε)/(ln|r|)

See that! The sign changes from < to > in my book and it says this is because ln|r| is negative as it's less than 1.

But isn't (ln ε) less than 1 as well, like stupendously close to zero actually!

Shouldn't the two cancel each other out the second you take the logarithm of

both sides?

Also, for 0 (the trivial case);

|r^n - 0| < ε

|r^n| < ε

n ln |r| < ln ε

------------------------

n > (ln ε) / (ln|r|)

-------------------------

n > (ln ε) / (ln|0|)

n > (ln ε) / (1)

n > ln ε

Is that the way to do it or is this:

|r^n - 0| < ε

|r^n| < ε

|0^n| < ε

0 < ε

The idea is to set N = (ln ε) / (ln|r|) and as long as n > N the proof should hold, I'm just a bit confused by this minus thing and how we're supposed to arrive at the answer i.e. because of the way everything is set up it shows that n > n, I don't see it.