Subsequences
Calculus
Level
1
If a sequence converges, does every subsequence of that sequence also converge?
No
Yes
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1 solution
Every infinite subsequence of a convergent sequence converges!
A convergent sequence has (1) infinitely many terms in the "neighborhood" of the value it converges to and (2) a finite amount of terms outside of that neighborhood (by the definition of convergence). Any infinite subsequence must eventually "run out" of the terms outside of the neighborhood and must start including (an infinite amount of) terms within the neighborhood, and thus the subsequence converges. I hope this helps!