This is the proof that was given in the lecture and the last step is not completely clear to me. Excuse some formal mistakes and my language, as I am from Europe and sometimes unfamiliar with the correct translation of the formalities.
Let $(a_n)_{n\in \mathbb{N}}$ be a Cauchy Sequence in $\mathbb{K}$ and $a$ and accumulation point. It follows that the sequence converges to $a$.
Proof: Let $\varepsilon$ be arbitrary but fixed. We choose $n_\varepsilon \in \mathbb{N} $ such that :
$$ |a_n- a_m| < \frac{\varepsilon}{2} \; \forall n,m > n_\varepsilon > $$
and we choose $m_\varepsilon > n_\varepsilon $ such that:
$$|a-a_{m_\varepsilon}| < \frac{\varepsilon}{2} $$
Now up unto this point, it is just definitions, the first one coming from the Cauchy Sequence and the second one from the accumulation point. But now without any explanation, she completed the proof with this inequality, which I can not follow:
$$ \forall n > m_\varepsilon : |a-a_n| \leq |a-a_{m_\varepsilon}| + |a_{m_\varepsilon} - a_n| < \varepsilon \Rightarrow $$
The sequence converges to a. q.e.d.
Now my question: Where does this term come from?
$$ \leq |a-a_{m_\varepsilon}| + |a_{m_\varepsilon} - a_n| < \varepsilon $$
She got the last step from the Triangle Inequality. The Triangle Inequality is:
$$|c + d| \le |c| + |d|$$ For all real numbers
So in this case she let $c = a - a_{m_\varepsilon}$ and let $d = a_{m_\varepsilon} - a_n$
And so we get $$|a - a_{m_\varepsilon} + a_{m_\varepsilon} - a_n| \le |a - a_{m_\varepsilon}| + |a{m_\varepsilon} - a_n|$$
Then the $a_{m_\varepsilon}$s cancel out on the left hand side, so we get:
$$|a - a_{n}| \le |a - a_{m_\varepsilon}| + |a_{m_\varepsilon} - a_n|$$
And because we let $|a - a_{m_\varepsilon}| < \varepsilon/2$ and $|a_{m_\varepsilon} - a_n| < \varepsilon/2$ we have:
$$|a - a_n| \le |a - a_{m_\varepsilon}| + |a_{m_\varepsilon} - a_n| < \frac\varepsilon2 + \frac\varepsilon2 = \varepsilon$$
I hope that helped.