I am a non-mathematician who is taking a self-help course on real analysis. I want to prove that a constant sequence $\{x_n\}_{n=1}^{\infty}$ is Cauchy. I know it is true since $|x_n-x_m|=0$ for all $n,m\geq0$, but I do not know how to write it down in an $\varepsilon$ proof.
Can someone help me?
Start by referring to the definition of Cauchy sequence:For all $\epsilon>0$ there exists $N$ such that for all $m,n>N$ it holds that $|x_n-x_m|< \epsilon$. To prove that, you start the proof by "Let $\epsilon >0$". Now you have a single $\epsilon$ and for it you need to find a suitable $N$. Based on what you state in the question, can you suggest a value for $N$ that will do that job?
There is also another way to prove this. You can refer to the fact that every convergent sequence is Cauchy and thus prove instead that a constant sequence converges. The approach will be similar, namely refer to the definition (i.e., spell it out) and correctly interpret what needs to be done.