Why does "$x$ multiple of $y$" make the subadditivity property non-strict?

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Why does "$x$ multiple of $y$" make the subadditivity property non-strict?

I tried plugging in $x=\lambda y$, but upon quickly viewing the L.H.S. of the triangle inequality. $$\|x+y\| \leq \|x\| +\|y\|$$ as $$\|\lambda y + y\|$$ I don't see where the equality comes from?