### Abstract:

The purpose of this study is an attempt to explore sufficient condition for the convergence of Riemann integrable functions on any positive linear functional. The Arzela dominated convergence theorem holds for the Riemann integral provided it is assumed that the limit function is Riemann integrable. Yet, one can see that a simple modification in the Arzela’s dominated convergence theorem shows that the theorem can be expressed in another form which holds true for a particularly defined positive linear functional of Riemann integrable function. Consequently, we need an extended suffice convergence condition which holds true for any positive linear functional under the frame work of compact Hausdorff space and for this we obtain a theorem developed by W.F.Eberlein. To this effect, this study shows how to prove the classical convergence theorem made by W.F.Eberlein using the facts and generalizations from Sequence in particular Subsequence and Cauchy Sequence, Lp Space and unifying some well known convergence theorems.
Key words: Convergence of Riemann integrable functions, Positive linear functional,
Arzela dominated convergence theorem, Convergence theorem by W.F.Eberlein.