Real Analysis – Shouldn’t the Harmonic Series Converge?

Question

If a sequence converges in a metric space, it is Cauchy, and in $\mathbb{R}^k$ every Cauchy sequence converges.

Therefore, in $\mathbb{R}^k$ a sequence converges iff it is Cauchy.

Let $\{s_n\}$ be a sequence in $\mathbb{R}$ where each $s_n=\sum_{k=1}^na_k$.
Therefore, by the above, every series converges iff
$$\left | \sum_{k=m}^n a_k\right| <\epsilon$$
For a given $\epsilon >0$ and an integer $N$ such that $N\le m\le n$. If $n=m$ then the statement reduces to:

A series converges if and only if
$$|a_n| < \epsilon $$
For a given $\epsilon >0$ and an integer $N$ such that $N\le n$.

This clearly cannot be (e.g Harmonic series). When does the equivalence become an implication.

Answer 1

In the definition of a Cauchy sequence, you have not gotten the order of the quantifiers correct. The partial sums of a series are Cauchy iff for all $\epsilon > 0$ there exists an $N$ such that for all $m,n \ge N$, $|\sum_{k=m}^{n} a_k| < \epsilon$. Notice the "for all $m,n$." You cannot just let $m=n$ and verify the condition for that case.