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homework4solnmat25

# homework4solnmat25 - Homework 4 Solutions 1 For each of the...

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Homework 4 Solutions 1. For each of the following sequences ( s n ), find the set S of subsequential limits, and use this to find lim inf s n and lim sup s n . (a) (11.1) s n = 3 + 2( - 1) n . Solution: ( s n ) = (5 , 1 , 5 , 1 , . . . ), so S = { 1 , 5 } , and therefore lim sup s n = 5 and lim inf s n = 1. (b) s n = n 2 ( - 1 + ( - 1) n ). Solution: ( s n ) = (0 , - 2 , 0 , - 18 , 0 , - 50 , . . . ), so S = {-∞ , 0 } , and there- fore lim sup s n = 0 and lim inf s n = -∞ . (c) ( s n ) = (0 , 1 , 2 , 0 , 1 , 3 , 0 , 1 , 4 , . . . ). Solution: S = { 0 , 1 , ∞} , so lim sup s n = and lim inf s n = 0. 2. Prove that every unbounded sequence contains a monotone subsequence that diverges to either or -∞ . Solution: If ( s n ) is unbounded above, for each choice of K , there exists a sequence element s n k so that s n k > K . The subsequence ( s n k ) clearly diverges to . Proof is similar if ( s n ) is unbounded below. 3. Suppose x > 1. Prove that lim x 1 /n = 1. Solution: First of all, x 1 /n is bounded below by 1 since x > 1. Note the quantity x 1 n ( n +1) > 1, so x 1 n +1 < x 1 n ( n +1) x 1 n +1 = x 1 n , which means the sequence is decreasing. Since it is decreasing and bounded below by 1, a limit exists for the sequence, call it s . Note any subsequence must converge to s . However, the subsequence x 1 / 2 n converges to

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homework4solnmat25 - Homework 4 Solutions 1 For each of the...

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