I need to show that if f is uniformly continuous on (a,b], then the right hand limit of f(x) exists.

This is what i have so far:

If f is uniformly continuous on (a,b], given epsilon >0, there is a delta > 0 s.t. |x-y| < delta -> |f(x)-f(y)| < epsilon for all x,y in (a,b]

Let x be in (a, a + delta).

|f(x) - f(y)|

<= |f(x) - L|+|L-f(y)|

< |f(x) - L|

< epsilon.

Thus, the right hand limit of f exist

Does this make sense?