Answer to Question #124186 in Real Analysis for Kurdish Ibrahim

Question

Q/Prove that 1+x<3^x for all x>0
Q/Prove that there is at least one x€R such that 3^x=3-x
Q/Suppose that lim n_infinity Sn=1 using definition ,prove that lim n to infinity (1+2Sn)=3
Q/(i). State the definition of a Nested Sequence of Sets and write an example. If {In}n€N is a nested sequence of nonempty closed bounded intervals, then Prove that
E=⋂In={x|x € In for all n€N}
contains at least one number. Moreover, if the lengths of these intervals satisfy as , then E contains exactly one number.
Q/(i). State Monotone Convergence Theorem or (MCT).
(ii).Prove that each of the following sequence converges to zero.Sn=(Sin n^4+n+1/n^2+1)/n
Q/prove that x-1/logx is uniformly continuous on(0,1)

Answer 1

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