1.I .3D. 3 \mathrm{D} \quad

Analysis I
Part IA, 2001

What does it mean to say that unlu_{n} \rightarrow l as nn \rightarrow \infty ?

Show that, if unlu_{n} \rightarrow l and vnkv_{n} \rightarrow k, then unvnlku_{n} v_{n} \rightarrow l k as nn \rightarrow \infty.

If further un0u_{n} \neq 0 for all nn and l0l \neq 0, show that 1/un1/l1 / u_{n} \rightarrow 1 / l as nn \rightarrow \infty.

Give an example to show that the non-vanishing of unu_{n} for all nn need not imply the non-vanishing of ll.