I'm trying to understand the proof for the lemma:
$\|\alpha _1 e_1 + \alpha _2 e_2 + \cdots + \alpha_n e_n\| \geq c (|\alpha_1|+|\alpha_2|+\cdots+|\alpha_n|)$ where $c>0$ and the $e_i$s are finite and linearly independent.
The proof I'm referring to is found in many places, like Kreyszig’s book and on page 16 of this PDF file; and in other locations.
The proof is based on proving that no sequence of the form $(\alpha_1 e_1 + \alpha _2 e_2 + \cdots + \alpha _n e_n)$ could converge to a sequence with zero norm.
What I don't quite get is the following:
Why the proof is not simply stated as: If the lemma is not true, then $\|\alpha _1 e_1 + \cdots + \alpha_n e_n\| = 0$ which is not possible since the $e_i$s are independent. I mean, why do they involve the sequence convergence business here?
Even when they assume that a sequence has to converge to a zero normed sequence, why does it have to be with the constraint that $|\alpha_1^{(m)}|+|\alpha_2^{(m)}|+\cdots+|\alpha_n^{(m)}| = 1$? Shouldn’t we consider that any possible sequence converges to zero normed sequence without the restriction that the sum should always be one?
Thanks a lot!