4. Prove that for any n∈Z+, An ≤Sn.

Expert Solution

Colby Messinger answered 2 years ago

Sn = (1+(1/n))^n (a) Prove Sn is strictly increasing (b) bounded below by 2 and above...

Sn = (1+(1/n))^n (a) Prove Sn is strictly increasing (b) bounded below by 2 and above by 3 (c) Sn converges to e (d) Obtain an expression for e (e) Prove e is irrational

Prove that Sn is generated by {(i i + 1) I 1 < i < n - 1}.

(a) Prove that Sn is generated by the elements in the set {(i i+1) : 1≤i≤n}....

(a) Prove that Sn is generated by the elements in the set {(i i+1) : 1≤i≤n}. [Hint: Consider conjugates, for example (2 3) (1 2) (2 3)−1.] (b) ProvethatSn isgeneratedbythetwoelements(12)and(123...n) for n ≥ 3. (c) Prove that H = 〈(1 3), (1 2 3 4)〉 is a proper subgroup of S4.

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Prove the following theorem: Theorem ∀n ∈ Z, n is either even or odd (but not both). Your proof must address the following points: 1. n is even or odd (and nothing else). 2. n is odd =⇒ n is not even (hint: contradiction). 3. n is even=⇒ n is not odd (hint: contrapositive). The first point is a bit more difficult. Start by making a statement about 0. Then assuming that n is even, what can you say about...

Question

4. Prove that for any n∈Z+, An ≤Sn.

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