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 said: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 \}##?
Here is the context of the solution: https://math.berkeley.edu/~talaska/old-104/hw06-sol.pdftnich said: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##?
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.Mr Davis 97 said: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##.
Clarifying a line in a proof means to provide a more detailed explanation or justification for a particular step in the proof. This can be done to make the proof more understandable or to address any potential confusion or questions that may arise.
Clarifying a line in a proof is important because it helps to ensure the accuracy and validity of the proof. It also allows others to follow the reasoning and understand the steps taken to reach the conclusion.
There are a few signs that a line in a proof may need clarification. These include unclear or vague language, an assumption that may not be obvious, or a step that seems to be missing in the logical progression of the proof.
Sure, for example, in a proof by induction, one may need to clarify a line by explicitly stating the base case and the inductive hypothesis before proceeding with the inductive step. This helps to make the proof more understandable and convincing.
To effectively clarify a line in a proof, one can use clear and concise language, provide necessary definitions or background information, and explain any assumptions or intermediate steps taken in the proof. It can also be helpful to provide visual aids or examples to make the clarification more concrete.