Prove that the language L = {w1, w1w2, w1w2w3, ..} is regular, provided wi is in a regular language
Your language is known as $L^+$.

It is a very standard property of regular languages that if $L$ is regular then so is $L^*$; this can be proved in several ways, and you can find proofs in textbooks and online sources. Since $L^+$ is either equal to $L^*$ or obtained from it by removing $\epsilon$ (the empty string), then $L^+$ is also regular.