I am reading Elementary Analysis by Kenneth Ross and am really struggling, especially with the choice of notation and why certain things are included in the proofs.
In chapter ten he is proving that for a sequence $s_n$, if the $\lim(s_n)$ is defined, then $\liminf(s_n)$ = $\lim(s_n)$.
In his proof he states: "Suppose $\lim(s_n) = +\infty$. Let $M$ be a positive real number. Then there is a positive integer $N$ so that $n>N$ implies $s_n > M$. Then $u_N$ (defined to be $\inf{s_n: n>N}$), $\geq M$. It follows that $m > N$ implies $u_m \geq M$. In other words, the sequence $(u_N)$ satisfies the condition defining $\lim u_N = \infty$."
I understand this is a silly question but why does he switch the index from $n$ to $m$. Why can't he say that this implies that $u_n > M$ for all $n > N$, and hence that the $\lim u_N = \infty$?
In what you're suggesting there is a clash of variable names; if we expand the definition we get $u_n = \inf\{s_n : n > N\}$, which doesn't really make sense because of how $n$ is being used for two different things (on the left it is a fixed number, on the right it is an indexing variable). The solution is to choose a new letter so that this doesn't happen. He could have chosen to use a different indexing variable in the definition of the $u$'s, or a different letter when using the $u$'s (as he did). Either is fine.
In fact, he's really just being careful to try to explain things. Once the $u$'s are defined we could forget about the indexing variable used in their definition, so writing $u_n$ isn't really wrong, but it has the potential to cause confusion and the author was trying to avoid that.