Is Every Closed Algebraic Set of Dimension $n$ Contained in a Closed Variety of Dimension $n+1$

Question

Let $V$ be an algebraic variety of dimension $m$ over an algebraically-closed field of characteristic $0$, and let $n<m$ and $U\subset V$ be a closed subset of $V$. Must there exist a subvariety $U\subset U'\subseteq V$ with $U'$ of dimension $n+1$? In the affine case, this is just asking if the solutions to a system of $m-n$ or more polynomial equations are all solutions to a system of $m-n-1$ equations giving a variety. It is foundational to algebraic geometry that a potentially reducible $U'$ exists. I can show the result using Lagrange interpolation for the projective case when $n=1$, but beyond that I am lost. I am not particularly learned in algebraic geometry.