Definition of null sequence I am using:
$(a_n) \to 0$ as $n \to \infty \iff$ given $\epsilon > 0$, there's $N$ such that $n > N \implies |a_n| < \epsilon.$
The sequences below are not null sequences because we can choose appropriate epsilons such that the absolute values of every term of a sequence after a certain point is greater or equal to epsilon.
a) $2, 1, 0, -0.1, 0.01, -0.001, 0.01, -0.001,\ldots, 0.01, -0.001,...\ldots$
b) $1, \frac12, 1, \frac14, 1, \frac18,\ldots$
The answers in my book(for proving the given sequences are not null) are $\epsilon = 0.01$ or less for (a) and $\epsilon = 1$ or less for (b).
I don't get these answers because in (a) if we choose $\epsilon = 0.01$, then there's always a term in the sequence such that $|-0.001| < \epsilon $ and in (b) if $\epsilon = 1$, then there's always a term $\frac 1n$ with $n \neq 0, 1$ such that $|\frac1n| < \epsilon$.
Please, explain why we choose these epsilons.