I've just started reading a book on C*-algebras and within a few pages
I've already encountered a number of "it is easy to show that"
statements which I can't figure out how to prove. Here are a few of
them, any solutions or hints would be welcome.

Firstly suppose that U is a C*-algebra without identity. We define a
new C*-algebra U_1 as follows: let the elements of U_1 be pairs (a, A)
with a a complex number and A in U. Addition and involution are
defined in the obvious way, and multiplication is defined by (a, A)(b,
B) = (ab, aB + bA + AB). A norm on U_1 is defined as follows:

||(a, A)|| = sup {||aC + AC|| | C in U, ||C|| = 1}.

My first problem is, how does one show that this norm satisfies the
product inequality, i.e.

||(a, A)(b, B)|| <= ||(a, A)|| ||(b, B)||?

Secondly, how does one show that this norm makes U_1 complete? I
assume that this is proved by first proving that if ((an, An)) is a
Cauchy sequence then the sequences (an) and (An) are Cauchy, but I
don't know how to show that this is the case - it seems likely that |
an| and ||An|| are bounded by some function of ||(an,An)|| but my
attempts to show this have drawn a blank.

Now suppose that U is a C*-subalgebra of a C*-algebra V and that V has
an identity 1. The book states that U_1 is identifiable as the
smallest C*-subalgebra of V containing U and 1, by which I assume it
means that the map

(a,A) |--> a1 + A

is an isomorphism between U_1 and the aforementioned smallest
subalgebra. I am stuck on how to show that this map preserves the
norm; it is easy to see that ||(a,A)|| <= ||a1 + A|| but I don't know
how to show the other inequality.