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This is actually only true when the yield curve is upward sloping. Intuitively, zero rates are average forward rates; e.g., the 10-year zero coupon yield is the geometric average of the 0y forward 1y rate, 1y forward 1y rate, 2y forward 1 year rate, ..., and 9y forward 1y rate: $$(1 + y_{10})^{10} = (1 + f_{0,1})(1 + f_{1,2})\ldots(1 + f_{9,10}).$$ So ...