can someone explain this proof please, I added a star to the inequalities I don't see/understand.

if | | is a norm on a field K and if there is a C > 0 so that for all integers n |n.1| is smaller than or equal to C, the norm is non archimedean (ie the strong triangle inequality is true)

proof: if x and y in K

[tex]\[

\begin{array}{l}

|x + y|^n \le \sum\limits_{k = 0}^n {|\frac{{n!}}{{k!(n - k)!}}} x^k y^{n - k} | \le *(n + 1).C.\max \left( {|x|,|y|} \right)^n \\

|x + y| \le *\mathop {\lim }\limits_{n \to \infty } \left[ {(n + 1)C.\max \left( {|x|,|y|} \right)^n } \right]^{1/n} * = \max \left( {|x|,|y|} \right) \\

\end{array}

\][/tex]

if | | is a norm on a field K and if there is a C > 0 so that for all integers n |n.1| is smaller than or equal to C, the norm is non archimedean (ie the strong triangle inequality is true)

proof: if x and y in K

[tex]\[

\begin{array}{l}

|x + y|^n \le \sum\limits_{k = 0}^n {|\frac{{n!}}{{k!(n - k)!}}} x^k y^{n - k} | \le *(n + 1).C.\max \left( {|x|,|y|} \right)^n \\

|x + y| \le *\mathop {\lim }\limits_{n \to \infty } \left[ {(n + 1)C.\max \left( {|x|,|y|} \right)^n } \right]^{1/n} * = \max \left( {|x|,|y|} \right) \\

\end{array}

\][/tex]

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