I am currently studying Section 2.4 Finite dimensional normed spaces in Kreyszig - Introduction to Functional Analysis.
In particular, Theorem 2.4-2 (Completeness): Every finite dimensional subspace $Y$ of a normed space $X$ is complete. In particular, every finite dimensional normed space is complete.
The proof of this theorem is clear to me, but I had an example in mind which seems to contradict the statement of the theorem.
If we consider $X = C[0,1]$ with the norm $||x|| = \sup_{t \in[0,1]} |x(t)|$ and define $$Y:=\{p:[0,1]\rightarrow\mathbb{R}:p(t) := \sum_{j=0}^n c_j t^j, n \leq N\}$$ for some fixed $N \in \mathbb{N}$.
Then $Y$ is a finite dimensional subspace of $X$ since $\{1,t,t^2,\ldots,t^N\}$ is a basis. However, the Cauchy sequence $(p_n)$ defined by $p_n(t) = \sum_{j=0}^n t^j/j!$ converges to the exponential function, which is not a polynomial, so not every Cauchy sequence converges, we conclude that $Y$ is not complete.
Where is my example false?