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dchilow

Member

Registered: 2007-03-05

Posts: 27

Help with Real analysis.

I need help with these practice problems. If anyone knows how to do them, I would be very appreciative if they would help.
{Xn} is a sequence.
1)  Suppose that {Xn} converges to L and that L > 0.  Prove that there exists a positive number E and a positive integer N such that Xn ≥ E for all n ≥ N.

2)  Let {Xn} be a sequence of real numbers that converges to infinity.

     (a)  Prove that {Xn} is bounded below.
     (b)  Let {An} be another sequence.  Suppose there is a positive integer N such that An ≥ Xn for all n ≥ N.  Prove that {An} converges to infinity.

4)  Let {An} and {Bn}  be two number sequences.  Suppose {An} is increasing, {Bn} is decreasing, and An≤ Bn for all n.

     (a)  Prove that both sequences converge.
     (b)  Must both sequences converge to a same limit?  Give proof or disproof.

5)  Let {An} be a sequence of nonnegative real numbers that converges to A.  Prove that the sequence {√An} converges to √A.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Help with Real analysis.

1.
Set ε (from the definition of convergence) equal to L/2, and try working from there.

2.
Something diverges to infinity if for all real M, there exists an N such that n > N ⇒ x_n > M.
Set M equal to (say) 1. Then by the above definition, an N exists such that n > N ⇒ x_n > 1.

Now consider the set {x_n| n ≤ N}. This is a finite set, and so has a minimum. If the minimum of that set is less than 1, then that is your lower bound. Otherwise, 1 is.

4a.
If a sequence is increasing and bounded above, then it is convergent. We're told that A_n is increasing, so to prove convergence we need to show that it is bounded above.

Now, since B_n is decreasing, B_n ≤ B_1, for all n. Hence, A_n ≤ B_1, for all n.
B_1 is therefore an upper bound for {A_n} and so {A_n} is convergent.

You can use a similar argument to prove the convergence of {B_n}.

4b.
The sequences don't necessarily converge to the same limit. It isn't too hard to find a counter-example.

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It wanted to be normal.