Gradient Descent and Learning Rate Assuming that our function is convex and only has a single minimum. Would gradient descent ever hit the minimum if the learning rate was extremely small? Or would it bounce back and forth without ever hitting the minimum. What would the case be if the function contained  two minimums? One that is the global and the other being the local.

Consider the convex function $f(x) = |x|$. Then gradient descent would bounce back and forth forever: since the gradient is always either $+1$ or $-1$, with learning rate $\lambda$ and if $x\in(0, \lambda)$, then $x$ would alternate between $x$ and $x-\lambda$.
If, as another example, $f(x) = x^2$, then gradient descent would converge to the minimum without attaining it. But you would not observe it because of the finite resolution of floats.
Note that there are various extensions of ordinary gradient descent, from more basic ones with decreasing learning rates up to more sophisticated ones like Adam or Nadam.
If there are several local minima, gradient descent can be caught in local minima without reaching the global one. But then your presumption of convexity would be violated.
