Show that every Cauchy sequence is bounded

Question

I am supposed to prove, that every Cauchy sequence is bounded.

From the definition of Cauchy sequence:$\left ( \forall \varepsilon > 0 \right )\left ( \exists n_{0} \in\mathbb{N} \right ) \left ( \forall m,n \in\mathbb{N}:m,n\geq n_{0} \right )d\left ( x_{n},x_{m} \right )< \varepsilon $.

We can fix $ \varepsilon$: $\varepsilon=1$. Then $m\geq n$, so that $d\left ( x_{n},x_{m} \right )< \varepsilon $. Let p be point in a space...

I do not know how to continue, or if it is correct. I want to somehow include triangle inequality, but I am not really sure, if that is correct.

Any help?

Answer 1

You are on the right track. I will use $|x-y| $ for $d(x,y)$ so $|x| =d(x,0)$.

As you said, for $\epsilon = 1$ we have $n_0$ s.t. for all $n,m\ge n_0$ we have $|x_n-x_m| <1$ in particular for all $n\ge n_0 , |x_n-x_{n_0}|<1$.

So , $|x_n| < |x_{n_0}|+1$ for $n\ge n_0$.

Do you see how to continue?