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__Problem statement__Given f:[1,2]->R defined by f(x) = x^2. Show that this function is continuous

__Problem Solution__*(my version at least)*1- It is known sequence {a[n]b[n]} converges to {ab}

2- Definition of continuous: if every sequence {cn} in f we have f(cn) -> f(c)

3- Given our domain of 1<= f <=2 assume our sequence an-> a=1 and bn-> b=2

4- Therefore f(anbn) = f(1*2) = f(a*b) = f(1*2) = 4

5- Therefore this function is continuous.

My concern is am I taking a leap of faith in 3,4 which is the essence of my proof.

Is there a better way to prove this or is this mathematically sufficient.

Thanks

Asif