I recently read the proof of why every convergent sequence is a Cauchy sequence and the mechanics of the proof make total sense to me. However, the proof is rather **different** from what my intuitive brain pictured. Let me explain. For a convergent sequence we know the sequence $\{ x\_n \}\_n$ is converging to a specific point say $x$. What this means to me intuitively is the following picture:

[![enter image description here](https://i.stack.imgur.com/0o3mD.png)](https://i.stack.imgur.com/0o3mD.png)

In other words, since the distance from $x\_n$ to $x\_m$ to $x$ is bounded then its "obvious" that the distance between any $x\_m$,$x\_n$ should also be bounded. But not only that it seems that the distance between any $x\_n$ and $x\_m$ should be bounded by some $\epsilon'$ less than the original $ \epsilon$ in the convergent definition. However, the proof seems to imply that it twice as big as the original $\epsilon$. Why is the relation between the epsilons the other way round from what I expect if the distance between $x\_n$ to $x\_m$ is included in the distance between $x\_n$ and $x$. Essentially since in the picture $ \| x\_n - x\_m \| \leq \| x\_n - x \| $, then its unexpected to have the constant bounding the Cauchy sequence be twice as large. 

Is the constant to loose or do I have a fundamental misconception of how the proof should have turned out?

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Maybe I'll try to explain the proof in my own words and reveal my misunderstanding? 

Consider a sequence $ \{ x\_n\}\_n$ that converges to $x$. This means that $\forall \epsilon\_{convergent} > 0$, $\exists N \in \mathbb{N}$ s.t. $ \| x\_n - x\| < \epsilon\_{convergent} $. Fix a $\epsilon\_{convergent}$. This corresponds to some $N$. Let it be $N\_{convergent}$. Then any two points in the sequence after $N\_{convergent}$ have the property $ \| x\_n - x\| < \epsilon\_{convergent}$, $ \| x\_m - x\| < \epsilon\_{convergent}$. Therefore consider the quantity of interest and apply the definition of norm (using triangle inequality):

$$ \| x\_n - x\_m \| = \| x\_n - x + x - x\_m \| \leq \| x\_n - x\| + \| x\_m - x \|$$

but since we know both are bounded by $\epsilon\_{convergent}$ we get that for that same $N\_{convergent}$ the above difference is bounded by $\epsilon\_{cauchy} = 2 \epsilon\_{convergent}$. Which we used all the rules and axioms correctly so mechanically it looks fine but the constant $\epsilon\_{cauchy}$ seems to be larger than I expected (when I thought it should have obviously been smaller), which could mean my intuition is wrong. I just hope to **correct it** or maybe I don't understand the proof as well as I thought.