Prove that A = B for: A = {(x,y) e Rº : +y/<1} B = {(z,y) € RP: (71+ y)² < 1}
4. Prove that SNS Here r < n and r < m.
(c) [5 points] Prove that f(r) [5 p ) = Σ (-1-rn oints Prove that f(x converges uniformly on [-c, c when 0<c<1. lenny
For all n E N prove that 0 <e- > < 2 k!“ (n + 1)! k=0 Hint: Think about Taylor approximations of the function e".
Let T be a bounded subset of R and let S CT. Prove that supS < supT.
Exercise 3. Suppose that |2 < 2. Prove that the series converges absolutely.
We write R+ for the set of positive real numbers. For any positive real number e, we write (-6, 6) = {x a real number : -e < x <e}. Prove that the intersection of all such intervals is the set containing zero, n (-e, e) = {0} EER+
PROVE: 4. If f : R → R is a strictly increasing function, f(0) = 0, a > 0 and b > 0, then
(b). Determine the truth value of " Vc e R+, In € Z+ <c", and justify your answer.
1. Let x, a € R. Prove that if a <a, then -a < x <a.