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Edit I've made a mistake in the formulation. There should be in inclusion, not an equality.

Let $E$ be a Banach space. Let $\varnothing\neq K_j\subset E$ be compact for all $j\ge1$ such that $K_{j+1}\subset\{x+y: x\in K_j~\mbox{and } y\in E~\mbox{ such that } \|y\|\le \eta_j\}$ where $(\eta_j)$ a sequence of strictly positive real numbers such that $\sum_j\eta_j$ converges in $\mathbb{R}$.

How can I show that $\cup_{j\ge1}K_j$ is relatively compact using total boundedness? I have been able to prove this using a diagonal argument, but the proof is quite messy. I feel like using total boundedness is easier since the closure of this union is complete.

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