Shouldn't the harmonic series converge? 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.
 A: If $\epsilon>0$ and $m\in\mathbb{N}$ then there is an $n\in\mathbb{N}$, $n>m$ such that
$$ \sum_{k=m}^n\frac{1}{k}>\epsilon $$
A: 
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 $N\le m\le n$.

What are $m$ and $n$ in this statement?  They have not been introduced.  This statement doesn't make sense.
Here's what you should have said.  

The following two statements are equivalent:
  
  
*
  
*The series $\sum_{k=1}^\infty a_k$ converges.
  
*If $\epsilon > 0$, then there exists a positive integer $N$ such that if $m$ and $n$ are positive integers and $N < m \leq n$, then $\left | \sum_{k=m}^n a_k \right| < \epsilon$.
  

A: 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.
A: It is true that a sequence in $\mathbb{R}$ converges if and only if it is Cauchy, and as such a series in $\mathbb{R}$ converges if and only if the sequence of partials sums $\{S_k\}_{k=0}^\infty$ is Cauchy.  The problem is that you are not using the definition of Cauchy correctly.  For a sequence $\{x_k\}_{k=0}^\infty$ it reads that for every $\epsilon >0$ there exists $K \ge 0$ such that if $m,n \ge K$, then $|x_n - x_m | < \epsilon$.  Here one does not get to restrict to $n = m+1$, one must check all possible values of $m,n \ge K$.  Note, though, that it is possible to restrict to $m \ge n \ge K$ in the definition.
Let's look at this for the harmonic series.  In this case $S_k = \sum_{m=1}^k 1/m$.  You are looking at $S_{k+1} - S_k = 1/(m+1)$, which indeed can be made arbitrarily small.  In fact, to use the Cauchy definition we must consider arbitrary $k \ge \ell$ to get 
$$
S_k - S_\ell = \sum_{m=\ell+1}^k \frac{1}{m},
$$
and it is these partial sums of arbitrary length that cannot be made small.
A: I think you're confusion is with the quantifiers. Your initial statements are correct, but to put everything in more specific terms the correct statement, with correct quantifiers is:
Let $\{s_n\}$ be a sequence in $\mathbb{R}$ where $s_n=\sum_{k=1}^na_k$. Then the series converges iff $\textit{for all } \epsilon > 0$ $\textit{there exists } N\in\mathbb{N}$ such that $\textit{for all } n\geq m > N$ we have that 
$$\left|\sum_m^na_k\right| < \epsilon$$
This does not hold for the harmonic series since $\sum_{k=N}^\infty\frac{1}{k}=\infty for all N$
A: It is true that a series converges if and only if the following is true:

For any $\epsilon > 0$, there is some $N$ such that
  $$\left\lvert \sum_{k=m}^n a_k\right\rvert < \epsilon$$
  whenever $N\leq m \leq n$.

The statement in the box above implies this one:

For any $\epsilon > 0$, there is some $N$ such that
  $$\lvert a_n \rvert < \epsilon$$
  whenever $N\leq n$.

The reason is that the statement in the first box says what happens
for all $m,n$ such that $N\leq m \leq n$,
including the pairs $m,n$ in which $m=n$.
On the other hand, the statement in the second box does not tell you
what happens for all $m,n$ such that $N\leq m \leq n$,
only what happens in a small subset of those cases.
Therefore you cannot use the statement in the second box to conclude
the statement in the first box.
A correct statement using the second box is,

A series $\{a_n\}$ converges only if for any $\epsilon > 0$, there is some $N$ such that
  $$\lvert a_n \rvert < \epsilon$$
  whenever $N\leq n$.

What you did was to write "if and only if" when you should
have written "only if".
