Example involving Uniform Continuity Question: Could someone give an example of a sequence of uniformly continuous real-valued functions on the reals such that they converge point-wise to a function that is continuous but not uniform continuous. 
My attempt so far: I managed to prove this is true in the case of uniform convergence, so I'm convinced there is an example. I considered triangles such that they were symmetrical on the y-axis such that they got taller and closer however this converges point-wise to 0 which is uniform continuous

I'm not convinced this is the simplest example (I certainly wouldn't want to make it explicit), but it was fun :)
Lines are uniformly continuous and quadratics are not. Also, continuous implies is uniformly continuous on a compact domain. In addition, it's not too hard to show that if you glue two uniformly continuous functions together, the result is uniformly continuous. 
Now create a function that approximates a quadratic by being identical near the vertex and at some point on each side breaking off into the tangent lines. Therefore, considering a sequence of such functions whose "breakoff points" have $x$-values that go to $\pm\infty$; these satisfy the premises and get the conclusion, so you're good.
