Inequality proof using Cauchy-Schwarz and Jensen's inequality

Question

I am trying to understand a proof of a simple lemma from this paper (lemma 3) which I paste here (refer to the 1st inequality, $H$ is a Hilbert space):

The first step in the proof is to use assumption no. 2 and Cauchy-Schwarz so that $\langle x_i,Te_k \rangle \leq c\|x_i\|$, then there is no longer dependence on $T$ so the $sup$ can be removed. But now, the proof seems to use the following step: $$\sum_i\gamma_{ik}\|x_i\|\leq\|\sum_i\gamma_{ik}x_i\|$$ which I fail to understand - it looks like the triangle inequality but the sign is reversed. (The next steps of the proofs are understandable).

What am I missing?

EDIT the last inequality I wrote is obviously wrong, so the proof is not using it, but something else which I don't understand

Answer 1

First write $\sum_{k=1}^{K} \sum_{i=1}^{n} \gamma_{ik} \langle x_i,Te_k \rangle$ as $\sum_{k=1}^{K} \langle \sum_{i=1}^{n} \gamma_{ik} x_i,Te_k \rangle$ before applying Cauchy - Schwartz.