An Important Fact about Subsequences | Real Analysis

The kth term of a subsequence is at least k terms into the original sequence. We talk about this ad nauseam in today's real analysis video lesson. If we say (a_n) is a sequence and (a_{n_k}) is a subsequence, then we can say n_k is greater than or equal to k. This is what we will be discussing. #RealAnalysis

The fact that the kth term of a subsequence is at least k terms along in the original sequence is a consequence of the definition of subsequence. The terms of a subsequence must be taken from the original sequence, and they must be taken in the relative order they appear in the original sequence. That is to say, we can have our subsequence go like a_1, a_3, a_5, but not like a_3, a_1, a_5, since going to a_1 after a_3 would be moving BACKWARD in the sequence. By definition, each term in a subsequence needs to be moving forward in the original sequence.

Intro to Subsequences:    • Intro to Subsequences | Real Analysis  
Sequence Converges if and only if all its Subsequences Do:    • Sequence Converges iff Every Subseque...  

Real Analysis Playlist:    • Proof: Product of Absolute Values is ...  

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