Minor but troublesome detail in the proof of the beta function convergence

Hi, I have a problem with a single but crucial step in the proof of convergence for the beta function, over real numbers. To summarise the proof, given the integral expression of the beta function, we split the integral into two intervals (0,1/2) and (1/2,1) and then prove convergence separately. To do this (for the first integral), we make use of an inequality: tx-1.(1 − t)y-1 ≤ tx-1 for 0 < t ≤ 1/2; x,y > 0. I cant see the truth of this inequality. For instance, if we let t = 1/2 and x = y = 1/2, we get (1/2)-1/2.(1/2)-1/2 = 2 > (1/2)-1/2 = 21/2 I am aware I must be overlooking something but I can't just figure out what.