I'm very unsure of myself with proofs so I wanted to know if my reasoning here is correct and if this proof is valid:
Given a sequence ${a_n}$ which converges and a sequence ${b_n }$ that is bounded such that for every n the following is true, $ b_{{n+1}}≤b_n+(a_{{n+1}}-a_n) $. Prove that $b_n$ converges.
Proof: Since $a_n$ converges we know that it is also bounded and monotonic, therefore: $|a_{{n+1}}-a_n |≤0$ for all $n$. (Otherwise $a_n$ would be unbounded and that’s contrary to the definition of a converging sequence)
If $b_n$ indeed converges, I would expect the following inequality to be true: $|b_{{n+1}}-b_n |≤0$ (The difference between consecutive terms must be decreasing and $0$ as $n$ reaches infinity)
Now I’m also given the inequality:
$b_{{n+1}}≤b_n+(a_{{n+1}}-a_n)$
$b_{{n+1}}-b_n
$|b_{{n+1}}-b_n |≤|a_{{n+1}}-a_n |$
$|b_{{n+1}}-b_n |≤|a_{{n+1}}-a_n |≤0 $ (according to my earlier statement about the definition of ${a_n}$.)
$|b_{{n+1}}-b_n |≤0$
According to the last inequality, the difference between the consecutive terms of $b_n$ are decreasing and eventually become zero as $n$ reaches infinity. Therefore $b_n$ converges. ∎