A convergent sequence converges to exactly one limit. That is, the limit of a sequence is unique. We'll prove this by contradiction in today's real analysis video lesson. We assume our convergent sequence converges to a and b, and that they are distinct, as in the limit is not unique. We then pick a sufficiently small epsilon and apply the definition of convergence to force the terms of our sequence to eventually satisfy two contradictory inequalities.
Real Analysis Playlist:https://www.youtube.com/playlist?list=PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli
Intro to Sequences: https://www.youtube.com/watch?v=YEiZWonJrOg&list=PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli&index=14
Definition of the Limit of a Sequence: https://www.youtube.com/watch?v=cTnlHZD5ss4&list=PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli&index=15
Proof (-1)^n Diverges: https://youtu.be/nEBF2oIS9Dk
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