Limit of x^n+y^n as n -> ∞: max(x, y)

[raphael3d]

show for two positive numbers x,y>0 that

limit for n->infinity : $$\sqrt[n]{x^n+y^n}$$ = max {x,y}

i don't know how to make a upper boundary(lower boundary is >0 i suppose)
something like assume for instance x bigger than y and than make a boundary with it, but how?

 

[mathman]

Assume x>y, take x outside the nth root. Use binomial theorem to show the limit of the nth root is 1.

 

[raphael3d]

x^n = (x^n+y^n) = x^n(1+(y^n)/(x^n))

=>

nth root of (x^n+y^n)=x*nth root of (1+(y^n)/(x^n)), and for x>y, =>x. the same applies for the opposite case.