- #1
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Homework Statement
1. Let xn and yn be sequences in R with yn+1 > yn > 0 for all natural numbers n and that yn→∞.
(a) Let m be a natural number. Show that for n > m
[tex]\frac{x_n}{y_n} = \frac{x_m}{y_n} + \frac{1}{y_n} \sum_{k=m+1}^{n} (x_k - x_{k-1})[/tex]
(b) Deduce from (a) or otherwise that
[tex]|\frac{x_n}{y_n}| \leq |\frac{x_m}{y_n}| + \sup_{k>m} | \frac{ x_k - x_{k-1} }{ y_k - y_{k-1} } |[/tex]
(c) Assuming [tex]\frac{x_n-x_{n-1}}{y_n-y_{n-1}} \to 0[/tex], show [itex]x_n/y_n \to 0[/itex].
(d) Assuming [tex]\frac{x_n-x_{n-1}}{y_n-y_{n-1}} \to L[/tex], show [itex]x_n/y_n \to L[/itex].
Homework Equations
N/A
The Attempt at a Solution
(a) was fine.
(b) Would the question be more correct to use sup k>m+1 instead, since a k-1 index is in the inside expression?
I'm not sure if this was the right thing to do as the question probably intended the sum to remain unsimplified, but I replaced it with xn-xm.
Using the triangle inequality and that y is strictly increasing so that yn-ym < yn, we get
[tex] | \frac{x_n}{y_n} | \leq | \frac{x_m}{y_n} | + | \frac{x_n-x_m}{y_n-y_m} | [/tex]
Then I'm not really sure what I can validly do after that.
(c) If (b) is assumed, then if we take the limit of both sides, it reduces to the statement that [tex] | \frac{x_n}{y_n} | - | \frac{x_m}{y_n} | \leq 0 [/tex] holds true for large n.
I don't think that's the right direction to go, certainly since it seems to imply x is decreasing when that was never given. Don't know what to do.
(d) No idea, but if I had (c) this one would probably be similar.