I was reading a proof on how any linear combination of compact operators is compact.
Let $U,V: X \to Y$ be compact linear operators and let $\alpha,\beta \in \mathbb{C}$. Then each bounded sequence $(x_n)$ in $X$ contains a subsequence $(x_{n(k)})$ such that $(Ax_{n(k)})$ and $(Bx_{n(k)})$ converge. Then the proof says that due to this $(\alpha A + \beta B)x_{n(k)}$ converges.
But what I dont understand is how we have the same indexing for the subsequence in the case of both operators. I would have thought that as $A$ and $B$ are two operators, the indexing for the subsequences could be different and that we should be considering, say, $(Ax_{n(k)})$ and $(Bx_{n(j)})$ as the convergent subsequences.
So why is this not the case and why does the same indexing apply to both operators when specifying the convergent subsequences?