Given sequences of positive real numbers $(\alpha_N)_{N=1}^\infty$ and $(\beta_N)_{N=1}^\infty,$ the notation $\alpha_N \lesssim \beta_N$ means that $\sup_N \alpha_N / \beta_N < \infty.$ Likewise, we write $\alpha_N \thickapprox \beta_N$ to mean $\alpha_N \lesssim \beta_N$ and $\beta_N \lesssim \alpha_N.$
If $\{e_j\}_{j=1}^\infty$ is a basis in $X$ and $\{f_j\}_{j=1}^\infty$ is a basis in $Y$. Then
how to prove that for every $N = 1, 2, 3, ...$
$$ \sup_{|A_1| + |A_2| = N} \left (\left \lVert \sum_{i \in A_1} e_i \right \rVert_X + \left \lVert \sum_{j \in A_2} f_j \right \rVert_Y \right ) \approx \max \{\displaystyle \sup_{|\Gamma| = N} \left \lVert \displaystyle \sum_{i \in \Gamma} e_i \right \rVert_X, \displaystyle \sup_{|\Gamma| = N} \left \lVert \displaystyle \sum_{j \in \Gamma} f_j \right \rVert_Y\}.$$
My attempt
Let $\alpha = \displaystyle \sup_{|A_1| + |A_2| = N} \left (\left \lVert \sum_{i \in A_1} e_i \right \rVert_X + \left \lVert \sum_{j \in A_2} f_j \right \rVert_Y \right)$
for every $\epsilon > 0$ there exists $A_1', A_2'$ s.t. $|A_1'| + |A_2'| = N$
satisfy
$\alpha - \epsilon < \left \lVert \displaystyle \sum_{i \in A_1'} e_i \right \rVert_X + \left \lVert \displaystyle \sum_{j \in A_2'} f_j \right \rVert_Y \leq \alpha.$
On the other hand, let $\displaystyle \sup_{|\Gamma| = N} \left \lVert \displaystyle \sum_{i \in \Gamma} e_i \right \rVert_X = Q$ and $\displaystyle \sup_{|\Gamma| = N} \left \lVert \displaystyle \sum_{j \in \Gamma} f_j \right \rVert_X = R.$
for every $\epsilon > 0$ there exists $\Gamma_\epsilon, \Gamma_\epsilon' : |\Gamma_\epsilon| = |\Gamma_\epsilon'| = N$ satisfy $$Q - \epsilon < \left \lVert \displaystyle \sum_{i \in \Gamma_\epsilon} e_i \right \rVert_X \leq Q.$$ and $$R - \epsilon < \left \lVert \displaystyle \sum_{j \in \Gamma_\epsilon'} f_j \right \rVert_Y \leq R.$$