Every sequence in a closed and bounded set S in sequence Rn has actually a convergent subsequence (which converges come a point in S).

You are watching: Every bounded sequence has a convergent subsequence

Proof: Every sequence in a closed and also bounded subset is bounded, therefore it has actually a convergent subsequence, i beg your pardon converges to a allude in the set because the collection is closed.

Conversely, every bounded sequence is in a closed and bounded set, therefore it has actually a convergent subsequence. AnotherBolzano-Weierstrasstheorem is:

Every bounded infinite collection of real numbers has at least one limit suggest or cluster point.

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Q1. The general solution the recurrence relation$$a_r-5a_r-1+6a_r-2=4^r,\ r\ge2$$is:
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