Saturday, 13 May 2017

real analysis - Convergence of Sequences (II)

Here are some other problems concerning converging of sequences:





Suppose p(x)=2x4+x2+3x+1 and q(x)=3x4+x3+2x+3. Define an=p(n)q(n) for nZ+. Prove that an23.




In other words, I want to show that (an23) is a null sequence. Now we know that an=2n4+n2+3n+13n4+n3+2n+3, nZ+



So an2n4+3n23n4=23+3n2



Thus an23. Is this the best way of doing the problem (i.e. it is bounded and monotonic increasing)?





Let [an,bn] be a nested sequence of closed intervals such that bnan0 and let (xn) be a sequence such that xn[an,bn] for all n. Prove that (xn) is convergent.




We know that (bnan)=0 for all n. So I think lim inf ([an,bn])=lim sup ([an,bn])



This is how I think of it: (xn) becomes "trapped" in smaller and smaller intervals until it is forced to converge to a point. Is this the right way to think about it?

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