given assumptions:{f_n} n=1 to infinty be a sequence of real-valued functions defined on [a,b] and suppose fn of functions converges uniformly on E to a function f which is defined on [a,b].fn is integrable on [a,b] then f is also integrable and limn-infnty[integral a to b fn(x)dx]=integral a to b(f(x)dx).now i am writing how i solved:since the sequence fn(x) is uniformly convergent to f i.e |fn-f|

Question

given assumptions:{f_n} n=1 to infinty be a sequence of real-valued functions defined on [a,b] and suppose fn of functions converges uniformly on E to a  function f which is defined on [a,b].fn is integrable on [a,b] then f is also integrable and limn->infnty[integral a to b fn(x)dx]=integral a to b(f(x)dx).now i am writing how i solved:since the sequence fn(x) is uniformly convergent to f  i.e  |fn-f| <epsilon then integral{a to x}(|fn-f|) <integral{a to x}(epsilon ) because of the absolute value  <=0 (left side) and  <epsilon(x-a) <=0 for the rhs since x in [a,b]now taking limit of both sid