Proportional allocation never does worse than SRSWOR of the same size

Consider the following sampling designs: a stratified sample scheme with SRSWOR within each stratum and a SRSWOR. Both have the same sample size. We are interested in estimating the population mean. Let $V^2_{\text{prop}}$ and $V^2_{\text{SRS}}$ be the corresponding variances of the usual estimators of the population mean. Then,