"Proving some subsets of regular sequences are regular..." was answered by students like you

fumefluosault7pa

Answered question

2022-02-12

Proving some subsets of regular sequences are regular sequences

stefjumnmt

Beginner2022-02-13Added 14 answers

Let us write

N = (x_{1}, \dots, x_{i}) M

, and for

j \geq i + 1

, let us write

I_{j} = (x_{i + 1, \dots, x_{j}}) \leq R

.
The following would be useful for you to check (2):

\frac{\frac{M}{N}}{I_{j}} (\frac{M}{N}) = \frac{\frac{M}{N}}{\frac{I_{j} M + N}{N}} \tilde{=} \frac{M}{I_{j} M + N} = \frac{M}{x_{1}, \dots, x_{j}} M

The case

j = n

in particular gives the following:

\frac{\frac{M}{N}}{I_{n}} (\frac{M}{N}) \tilde{=} \frac{M}{x_{1}, \dots, x_{n}} M

which would be useful for you to check (1).

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