Statement of problem: For f $\in B_1$ prove that there is a sequence $(g_n)$ in $C[0,1]$ s.t. $(g_n)$ converges to f pointwise on [0,1] and for each n we have $||g_n|| \le ||f||$. $B_1$ is the set of all bounded real valued functions on [0,1] which are a pointwise limit of continuous functions. Also ||f|| defined as $sup_{t\in [0,1]} |f(t)| $ i.e. the sup norm.
Context: Real Analysis by Carothers and Principles of Analysis by Rudin This was given as extra credit on a previous homework (already past due) and I would like to see how to tackle this problem to get ready for finals.
Work so far: We know that if f $\in B_1$ that it is continuous at on a dense set of points (Baire-Osgood Theorem). Also by definition we know there is a sequence of functions in $(f_n) \in C[0,1]$ converging to f $\in B_1$. My method of attack would be to prove by construction by taking this sequence and taking a subsequence for which the inequality applies but I am having a hard time justifying the viability of such a construction.
I'd appreciate any insight or help. Thank you.