I thought I completely understood uniform convergence until I ran into this statement that I cannot prove please help!

Suppose $\displaystyle \{f_n\}$ converges point wise on $\displaystyle [a,b]$ and for all $\displaystyle x\in [a,b]$ theres an open interval $\displaystyle I_x$ that contains $\displaystyle x$ such that $\displaystyle \{f_n\}$ uniformly converges on $\displaystyle I_x\cap [a,b]$. Show $\displaystyle \{f_n\}$ uniformly converges on $\displaystyle [a,b]$.

Can anyone walk me through this? Thank you!