Muntz-Szasz theorem

Let $\left\{ \alpha_{n}\right\}_{1}^{\infty }$ be a strictly increasing sequence of positive numbers with the property $\alpha_{n} \rightarrow 1$. Prove that the system $\left\{ t^{\alpha_{n}}\right\}_{n = 1}^{\infty }$ is complete

(H. Müntz, 1914). The system of powers $\left\{ t^{n_{k}}\right\}_{k = 0}^{\infty }$, where $n_{0} < n_{1} < \ldots$, is complete in the space $C[0,1]$ if and only if $n_{0} = 0$ and $\sum_{k = 1}^{\infty } \frac{1}{n_{k}} = \infty$.

Let $\lambda_{1}, \lambda_{2}, \lambda_{3}, \ldots , \lambda_{n}, \ldots$ be distinct positive numbers such that the series

is divergent. If a function $h(t)$ which is properly integrable over the interval $0\leq t\leq 1$ satisfies the condition

then $h(t)=0$ at every point of continuity $t$. (A generalization of Problem 139 of Part II.)

Suppose $\lambda_{1} > \lambda_{2} > \cdots$, and $\lambda_{n} \rightarrow 0$ in the Müntz-Szasz theorem. What is the conclusion of the theorem, under these conditions?

Prove an analogue of the Müntz-Szasz theorem, with $L^{2}(I)$ in place of $C(I)$.