Let f and g be differentiable on R such that f(1) = g(1), and f'(x) < '() for all r ER. Prove that f(x) = g(2) for 3 >1.
1 4.6.3. (Harder!) Let 0 < a < 1. Prove that for any n EN, (1 – a)” < 1+n·a
Let X be a continuous random variable. Prove that: P(21-; < X < xạ) = 1 - a.
Problem 10(20). Let x and y be vectors in R". Prove that |x"y| < ||x|||y- No work, no credit, messy work, no credit, missed steps, no credit disorganized work, no credit.
Let T be a bounded subset of R and let S CT. Prove that supS < supT.
let a,b > 0 . Prove that DI < Val
Prove AB _ BC__ AC aven. DEEF DE Prove: <A><D
Problem 10. Let f,g: [a,b] -R be Riemann integrable functions such that f(x) < g(x) for all x E [a,b]. Prove that g(x)
an (4) Let F be ordered field. Prove that the statement Vo: ZX\x6F*XWye FtX &<y)->(<y') is true (hist: Factor Y-X out of yº-x")
It is known that f :(0,2) + R is a differentiable function such that \f'(x) < 5 for all x € (0,2). Now let bn := f(2 – †) for all n € N. Prove that this is a Cauchy sequence.