- #1
Felafel
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Homework Statement
Let an be a bounded sequence and bn such that
the limit bn as n→∞ is b and
0<bn ≤ 1/2 (bn-1)
Prove that if:
an+1 ≥ an - bn,
then
lim an
n→∞
Homework Equations
The Attempt at a Solution
no clue :(
A sequence is a list of numbers that follow a specific pattern or rule. Each number in the sequence is called a term, and the number of terms in the sequence is infinite.
A convergent sequence is one that has a limit, meaning that the terms in the sequence approach a single, finite value as the sequence continues. A divergent sequence, on the other hand, does not have a limit and the terms in the sequence either approach infinity or oscillate between different values.
The existence of a limit in a sequence is determined by examining the behavior of the terms in the sequence as the index (or position) of the terms increases. If the terms approach a single, finite value, then the sequence has a limit. If the terms do not approach a single value, the sequence does not have a limit.
No, a sequence can only have one limit. If a sequence has multiple limits, it is considered divergent.
The limit of a sequence can be calculated by examining the behavior of the terms in the sequence and using mathematical methods such as the squeeze theorem, the monotone convergence theorem, or the ratio test. In some cases, the limit may be easily determined by simply looking at the pattern of the terms in the sequence.