Why is boundedness necessary in this proof that addition and scalar multiplication are continuous maps in the metric space induced by a norm?
Suppose that $\lambda _n \to \lambda $, $\mu _n \to \mu $, $\|x_n\| \to \|x\|$ and $\|y_n\| \to \|y\|$.
I'm confused by the following statement:
Noting that $\{||x _n |\} $ and $\{||y _n |\} $ are bounded, and using property (ii) and (iii) of the norm, it is easily verified that
$$\|(\lambda _n x _n +_mu _n y _n )-(\lambda x + \mu y )\|\to 0 $$
This seams to be a trivial fact from the triangle inequlity and absolut homogeniety of the norm. (And of course a convergent sequence is bounded.) Where do I need the boundedness?
Thanks in advance!