= Solution
\b[Additional observations weakly decrease the <posterior variance> at every fixed prediction input.] Apply part (ii) repeatedly to add observations one at a time. An observation whose remaining <conditional covariance> with $f_*$ is nonzero strictly decreases that <posterior variance>; one with zero remaining <conditional covariance> leaves it unchanged. Large observation noise reduces the information gained through the denominator $v_{n-1}(x_n)+\sigma^2$.
For fixed <Gaussian process> hyperparameters and a <Gaussian likelihood>, the <posterior variance> depends on the observation locations and noise level, but not on the observed numerical outputs. Hence the reduction holds for each possible new output, rather than only on average. The <predictive variance> of a future noisy observation also decreases by the same amount because its extra independent noise <variance> is fixed. Refitting the hyperparameters after adding data can change the <covariance function>; the monotonicity statement compares conditioning under one fixed model.
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