valid proof of geometry for inner product of two vectors

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I don't follow the last implication here: [![notes][1]]

How does for all $t \in [0,1]$ $$ \frac{t}{2} ||y||^2 + x^Ty \geq 0 $$ imply that $ x^Ty \geq 0$? [1]: https://i.stack.imgur.com/IJM8o.jpg