Problem on normed space. Let $X$ be a normed space, $Y$ a dense subspace of $X$ and $Z$ a closed finite-codimensional subspace of $X$. Is $Z\cap Y $dense in $Z$ ?
I have no idea how to solve this problem. I am using this website for the first time, any help would be appreciated.

yes, it is dense in $Z$.
draw a picture with one disk and a secant line, you will find the secret.
the line is a finite dimension space, there are many points belong to $Y$
around the line but not in the line, because the line-finite dimension space is nowhere dense(suppose $X$ is infinite dimensional)
