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Sequences Limit Consequences

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(a) Prove this operation:
Let {xn} and {yn} be convergent sequences.
The sequence{zn} where zn:=xn-yn converges and lim (xn-yn)=lim zn=lim xn-limyn

What I attempted was this:

Suppose {xn} and {yn} are convergent sequences and write zn:=xn-yn. Let x:=lim xn, y:=lim yn and z:=x-y
Let epsilon>0 be given. Find M1 s.t. for all n =>M1 we have |xn-x|<epsilon/2. Find M2 s.t. for n => M2 we have |yn-y|<epsilon/2
Take M:=max{M1,M2} For all n => m We have, |zn-z|=|(xn-yn)-(x-y)|

This is the part I get stuck at. I think I want to ultimately show the |zn-z|< epsilon, but I don't know if I am pursuing this correctly.

(b) Let xn:=[(n-1)(-1)^n]/n, find the lim sup xn and lim inf xn.

I tried doing the squeeze lemma, but that didn't work.

Thank you.

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Solution Summary

The expert proves that the subtraction of two convergent sequences is also convergent. Also, determining the liminf and limsup for a given sequence. Squeeze lemmas are examined.

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