Task:
Show that there exist binary codes that can correct one error and that have rate arbitrarily close to $1$.
This is asking for an existence proof, so either by contruction or using some well-known result, but I do not know where to even start with this. The statement seems so simple but one word is throwing me off: "arbitrarily close", which means that we can make the rate as close to $1$ as we like. We would have to come up with some code that has better and better rate if we make it larger, maybe the rate would be some fraction of the form $\frac{a-1}{a}$ for some natural number $a$. I think this corresponds to puncturing so-called "convolutional code"
Does anybody have a hint or tip?
My suggestion would be to look at Hamming codes of increasing length to see the rate converging to $1$.