Let $n\in \mathbb N$ and: $$ S_n = \sum_{k=1}^n x_n $$ Given $\{S_n\}$ is a bounded sequence prove that $\{x_n\}$ is a bounded sequence.
Similar questions have been asked here several times but all of them are in the context of series. I'm looking for a precalculus proof of boundedness of $x_n$.
Consider the following cases for $n$:
$$ S_1 = x_1 \\ S_2 = x_1 + x_2 \\ \dots \\ S_n = x_1 + x_2 + \dots + x_n $$
For some given $n$ we know that $S_n$ is a number. It gives that: $$ |S_1| = M_1\\ |S_2| = M_2 \\ \dots \\ |S_n| = M_n $$
Now since $\{S_n\}$ is bounded by some value it gives that: $$ \{ \forall k \le n: |S_k| \le M = \max\{M_1, M_2, \dots,M_n\} \} \\ $$ Now take $S_n$ and $S_{n-1}$ and subtract them:
$$ |S_{n} - S_{n-1}| = |x_n| $$
Since both $S_n$ and $S_{n-1}$ are less than or equal to some $M$: $$ |S_n - S_{n-1}| = |x_n| = |S_n + (-S_{n-1})| \le |S_n| + |-S_{n-1}| \le 2M $$
Therefore $|x_n| < 2M$ and hence bounded.
Questions:
- Is my proof valid?
- I feel like i have mixed precalculus concepts with concepts related to series (which are mostly considered in Calculus II), have I?