I Clarification of a line in a proof

[Mr Davis 97]

This comes from a line of a proof in my book, and I need help resolving why the equality is true. Suppose that $M>N$. Why is it true that $\displaystyle \sup \{\frac{1}{n} (s_{N+1} + \cdots + s_n) ~|~ n>M \} = \frac{n-N}{n}\sup \{s_n ~|~ n > N \}$?

 

[tnich]

This comes from a line of a proof in my book, and I need help resolving why the equality is true. Suppose that $M>N$. Why is it true that $\displaystyle \sup \{\frac{1}{n} (s_{N+1} + \cdots + s_n) ~|~ n>M \} = \frac{n-N}{n}\sup \{s_n ~|~ n > N \}$?

It's hard to tell without seeing what came before. You can set up a string of inequalities to show that $\displaystyle \sup \{\frac{1}{n} (s_{N+1} + \cdots + s_n) ~|~ n>M \} \leq \frac{n-N}{n}\sup \{s_n ~|~ n > N \}$. Do you have any information to show $\geq$?

 

[Mr Davis 97]

It's hard to tell without seeing what came before. You can set up a string of inequalities to show that $\displaystyle \sup \{\frac{1}{n} (s_{N+1} + \cdots + s_n) ~|~ n>M \} \leq \frac{n-N}{n}\sup \{s_n ~|~ n > N \}$. Do you have any information to show $\geq$?

Here is the context of the solution: https://math.berkeley.edu/~talaska/old-104/hw06-sol.pdf

My issue is in the bottom half of the first page.

For further context, the problem is

Let $(s_n)$ be a sequence of nonnegative numbers, and for each $n$ define $\sigma_n = 1/n(s_1 + s_2 + · · · + s_n)$. Show $\lim \inf s_n \le \lim \inf \sigma_n \le \lim \sup \sigma_n \le \lim \sup s_n$.

 

[tnich]

Here is the context of the solution: https://math.berkeley.edu/~talaska/old-104/hw06-sol.pdf

My issue is in the bottom half of the first page.

For further context, the problem is

Let $(s_n)$ be a sequence of nonnegative numbers, and for each $n$ define $\sigma_n = 1/n(s_1 + s_2 + · · · + s_n)$. Show $\lim \inf s_n \le \lim \inf \sigma_n \le \lim \sup \sigma_n \le \lim \sup s_n$.

I think the = sign is an error. It should be $\leq$. It doesn't invalidate the proof because it is in a string of inequalities anyway.