Proof that multiplying by the scalar 1 does not change the vector in a normed vector space.
Solution 1:
Nice question, this took me more thought than I expected.
Consider $a\in V$. Note that $0=\|1a-1a\|=\|1a-(1)(1a)\|=\|(1)(a-1a)\|=\|a-1a\|$.
Thus $a=1a$.