- #1

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- TL;DR Summary
- I want to prove that ##U_1 + U_2 + ... + U_k = span (U_1 \cup U_2 \cup ... \cup U_k)## by induction

__1) Base case ##k=2##__

##U_1 + U_2 = span (U_1 \cup U_2)##, which I understand how to prove is OK.

__2) Induction hypothesis__

We assume that the following statement holds

$$U_1 + U_2 + ... + U_{k-1} = span (U_1 \cup U_2 \cup \ ... \ \cup U_{k-1})$$

__3) Induction step__

$$U_1 + U_2 + ... + U_k = \underbrace{U_1 + U_2 + ... + U_{k-1}}_{=span (U_1 \cup U_2 \cup \ ... \ \cup U_{k-1})} + U_k = \ ? $$

But my issue is that I do not see how to move from ##span (U_1 \cup U_2 \cup \ ... \ \cup U_{k-1})+U_k##

Might you guide me on how to proceed further?

Thanks!