what you have learned about Proving Theorems
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what you have learned about Proving Theorems
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Ultimately, you should hopefully be able to prove the vast majority of the mathematics you know.
You do not necessarily need to prove things as you learn them, nor is it necessary or of benefit to do so. Oftentimes you will find that the proof for what you are learning is of a greater level and complexity than you can handle.
But as you learn new mathematics, the goal of a good mathematics curriculum is that you begin to understand the nuances that underpin basic math.
You begin to notice things, like the derivative of the volume of a sphere w.r.t. its radius is the surface area.
As you take further and further courses, into analysis, for example. In single-variable calculus, you might be taught about Taylor expansions, with some of the finer points about error estimates and derivations brushed under the carpet. Once you get to analysis you should be able to derive Taylor series.