Let $f_n :[a,b] \to R$ be a sequence of Riemann-integrable functions that converges pointwise to a Riemann-integrable function $f$. Suppose there exists a positive number $M$ such that $\left|f_n(x)\right| < M $ for every $n$ and $x$. I would like to prove that
$ \mathop {\lim }\limits_{n \to \infty } \int\limits_a^b {f_n \left( x \right)dx = } \int\limits_a^b {f\left( x \right)dx } $
Remark: Thanks for talking me about the "Dominated convergence theorem " but I don´t know, how to prove it )= , and other thing I do not know measure theory, this problem is from a calculus course, did not even know the definition of Lebesgue integral, and I can not use it for the problem )=
EDITED: Because I writed my question wrong. Thanks and sorry for tell me.