I'm trying to prove the following lemma for a vector space:
Suppose $X = span\{ e_1, ..., e_n \}$, with the $e_i$ linearly independent. (i.e. a basis, so you can uniquely write each $x\in X$ as a linear combination $\sum_j x^j e_j$ of the basis elements).
There exists $C > 0$ such that $|| x || \leq 1$ implies $| x^j | \leq C$ for all $j = 1, ..., n$.
At first I was thinking of using induction, since the base case is very simple : $||x^1e_1||\leq 1\implies |x^1|\leq 1/||e_1||$, but the inductive step looks much more difficult. We have $||\Sigma_{j=1}^n x^je_j|| \leq 1$, and using the triangle inequality $||x+y||\leq ||x||+||y||$, we have $||\Sigma_{j=1}^{n-1} x^je_j|| + ||x^ne_n||$, but I don't know how to show that $||\Sigma_{j=1}^{n-1} x^je_j|| \leq 1$ in order to make use of the inductive hypothesis. Any ideas?