Let $(c_0,\|\space.\|_{\infty})$ be the space of sequences $(x_k)_{k=0}^{\infty}$ such that $x_k\in{}\mathbb{K}$ and $x_k\rightarrow{}0$. I want to show that this space is complete.
I understand this requires me to take a Cauchy sequence in $c_0$ and showing it not only converges, but converges to a limit point (a sequence) in $c_0$.
I am comfortable with the definitions, but I am not sure how to approach the proof. This is how I tried to break down the problem:
Clearly a Cauchy sequence is bounded and I believe it does converge but can't show why (I can easily show that a convergence implies Cauchy).
After this, knowing it converges, I must show it converges to a sequence that itself converges to 0.
I did find a similar question someone else has posted but didn't find the solution very helpful.