Show that if $E$ is a finite dimensional subspace of $l_p$, there exists an integer $m$ such that $$\| P_m(x) \| \geq (1 - \dfrac{1}{n}) \| x \|$$ where $x \in E$ and $P_m$ is a projection map from $l_p$ onto a subspace generated by its first $m$ basis vectors.
The question comes from the proof of Proposition $7.1$. I have a problem in choosing integer $m$. Can anyone give some hint?
Note that $P_m(x)= \sum_{k=1}^m{a_kx_k}$ where $\{ x_k \}_{1 \leq k \leq m}$ is a basis vectors for $P_m(E)$.
Since $E$ is finite dimensional ($n$-dimensional, as it turns out when looking up the paper you are citing) there is a finite basis of unit vectors $v_1\ldots v_n$ of $E$. Each of these vectors can be arbitrariliy good approximated by any given basis of $l_p$. Assume such a basis has been chosen in advance, $(e_i)_{i\in \mathbb{N}}$, say.
Then, to any $\varepsilon > 0$ and $1\le i\le n$ there is a natural number $m_i$ and real (or complex) numbers $a_i^k$ such that $||v_i-\sum_{k=1}^{m_i} a_i^k e_k|| < \varepsilon $.
Now let $m:=max\{m_i:1\le i\le n\}$. Then in the space $l_p^{(m)}$ spanned by the first $m$ vectors $e_i$ each vector in the basis $v_i$ can be approximated up to $\varepsilon $ by vectors chosen from $l_p^{(m)}$. It is now a matter of choosing $\varepsilon$ and applying the triangle inequality (which I leave to you) to prove your claim.