The classical theorem of Müntz and Szász says that if \(\Lambda:= \{\lambda_ i\}^ \infty_{i=0}\), \(0=\lambda_ 0<\lambda_ 1<\cdots\) then \(M(\Lambda):=\text{span}\{x^{\lambda_ 0},x^{\lambda_ 1},\dots\}\) is dense in \(C[0,1]\) in the uniform norm if and only if \(\sum^ \infty_{i=1} \lambda^{-1}_ i=\infty\). This paper proves that if \(\Lambda\) is lacunary, that is \(\inf\{\lambda_{i+1}/\lambda_ i: i\in\mathbb{N}\}>1\) and \(A\subset [0,1]\) is of positive Lebesgue measure then \(M(\Lambda)\) fails to be dense in \(C[A]\) in the uniform norm. The key to the proof is the establishment of a bounded Remez-type inequality for lacunary Müntz spaces. This states that there is a constant \(c\) depending only on \(\varepsilon\) and \(\widetilde\lambda:=\inf\{\lambda_{i+1}/\lambda_ i: i\in\mathbb{N}\}>1\) (and not on \(n\) and \(A\)) so that \(| p(0)|\leq c\sup_{x\in A}| p(x)|\) for every \(p\in M(\Lambda)\) and \(A\subset [0,1]\) of Lebesgue measure at least \(\varepsilon>0\).