to prove infinity norm is indeed norm

Question

to prove infinity norm of a function which is equal to supremum of absolute value of that function, is indeed a norm.

I have a clue to check for the three axioms of norm. but tell me hoe to proceed

Answer 1

$1$: $\|f\|_\infty=0\iff f(x)=0$ $$ \begin{align} \sup_x|f(x)|=0 &\implies0\le|f(x)|\le0\\ &\implies f(x)=0 \end{align} $$ The other direction is self-explanatory.

$2$: $\|af\|_\infty=|a|\|f\|_\infty$ $$ \begin{align} \|af\|_\infty &=\sup_x|af(x)|\\ &=|a|\sup_x|f(x)|\\ &=|a|\|f\|_\infty \end{align} $$

$3$: $\|f+g\|_\infty\le\|f\|_\infty+\|g\|_\infty$ $$ \begin{align} \|f+g\|_\infty &=\sup_{\substack{x,y\\x=y}}|f(x)+g(y)|\\ &\le\sup_{\substack{x,y\\x=y}}|f(x)|+|g(y)|\\ &\le\sup_{x,y}|f(x)|+|g(y)|\\ &=\|f\|_\infty+\|g\|_\infty \end{align} $$