I'm learning multivariable calculus now, and when I look at the proof of the theorems on youtube, I always hear these sentences 'This proof is not rigorous enough, 'I'm being here a bit handwavy'.
For example, look at the proof of the arc length of a function: calculus textbook use Pythagoras equation like this $$ds^2=dx^2+dy^2$$ and then $$ds=\sqrt{dx^2+dy^2}=\sqrt{1+\frac{dy^2}{dx^2}}dx$$
integrating $$S=\int_a^b \sqrt{1+f'(x)^2}dx$$ and there is a similar method for finding the arclength of a vector valued function.
Now, since I'm gonna learn real analysis later, should I pay attention to the proofs of calculus, I think that this is just a waste of time since the proofs aren't rigorous enough?