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Is it not correct that the sequence converges to the zero function on $[0,1]$ which is in $C[0,1]$ in the $L^1$ sense, that is: the area between the $n$th monomial and the $0$ function goes to $0$ as $n \to \infty$?

My prof gave it as a proof that the normed space $C[0,1]$ with $L^1$ norm is not complete, but I don't get why because there exists a limit in $C[0,1]$.

suckling pig
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Alejandro
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    Isn’t it true that the limiting area is 0 using the 0 function ? – Alejandro Oct 25 '24 at 16:28
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    You are indeed correct, this sequence does converge to the zero function in the $L^1$-norm, so it's not a valid example. See this question for valid examples – Lorago Oct 25 '24 at 16:32

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