(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

1. Let x_{n}and y_{n}be sequences in R with y_{n+1}> y_{n}> 0 for all natural numbers n and that y_{n}→∞.

(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].

2. Relevant equations

N/A

3. 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 x_{n}-x_{m}.

Using the triangle inequality and that y is strictly increasing so that y_{n}-y_{m}< y_{n}, 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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Real Analysis: Stolz–Cesàro Proof

**Physics Forums | Science Articles, Homework Help, Discussion**