Does the set of all finite sequnces contain infinite sequences? In other words the set of all $(a_{i})_{i=1..N}$ . It looks like we can construct any infinite sequence since if we want to go one index ahead we just pick the next finite etc
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1Just read the first sentence of your question... – Asaf Karagila Nov 05 '19 at 08:26
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Or, read your username. – Eric Wofsey Nov 05 '19 at 16:49
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No. The set of all finite sequences contains arbitrarily long sequences. (This means for any length of sequence an adversary demands, you can exhibit a sequence of that length in the set.) It does not contain infinitely long sequences because, as it is defined, it does not.
Eric Towers
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