In Sheldon Axler's *"Linear Algebra Done Right" - 2$^{\textrm{nd}}$ Edition*, on the section for **Direct Sums**, the following proposition is stated.

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![](https://i.stack.imgur.com/nRls7.png)

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Following this is the proof of this 'if and only if' proposition - first in the forward direction ($V=U\_1 \oplus \cdots \oplus U\_n \,\implies \,$ (a) & (b) ). This is the part I am having trouble understanding.

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![enter image description here](https://i.stack.imgur.com/N49Gi.png)

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**Here is my problem:** To prove (b), the author seemed to simply use (b), thus proving it using itself, which is a tautology! How can the author say *"Then each $u\_j$ must be 0*, and then justify that with the "uniqueness part of the definition", which is exactly what the author is trying to prove? Have I misunderstood something?