tests for FLRW-cosmologies

Is there an observational test of verifying that we live in a FLRW-universe, i.e. of all (which?) symmetry assumptions behind choosing the Robertson-Walker-metric? Is this a sufficient condition? physics

bonus question: Can we test if the cosmological fluids are ideal?

maths bonus question: Can you prove (by induction) the Bernoulli inequality, namely that \((1+x)^n\geq 1+nx\) for all real numbers \(x\geq-1\)?