I recently came across a problem in a Numerical Algorithms (Computer Science) class that asked me to prove the Trapezoidal rule of integration.
Derive the trapezoidal rule for numerical integration which goes somewhat like: $$\int_a^bf(x)dx \approx \sum_{k=0}^{n-1} \frac{1}{2}h[f(x_k) + f(x_{k+1})]$$ for discrete values of $x_k$ in the interval $[a,b]$.
It seems to me that $\approx$ is not mathematically defined, thus the trapezoidal rule is formally ambiguous, thus there being nothing to prove. If I look beyond formal mathematics and think about what I was "actually" asked to prove, maybe I was being asked to show that the limit of the Riemann sum is integration, and if the limit is not taken, then that's the approximation. But here also the problem is that "approximation" is not well defined.
I would have gone ahead with a hand-wavy proof since this is a Computer Science class, but professor explicitly asked me to prove this formally without any hand-waving.
The appropriate solution to me seems what I mentioned above - since $\approx$ is not well defined, the statement to be proved is not well defined, thus it cannot be proved.
Please let me know whether this is formally (mathematically speaking) the right answer or the wrong answer.
As you say, there is no formal and exact definition for $\approx$, so if you want to prove something involving $\approx$, you must first make the $\approx$ more precise, and then prove your more precise statement. In the case of the trapezoidal rule, for example, you could assume that $f$ is $C^2$ on $[a,b]$ and then prove a bound of the form $$ \left|\int_a^b f(x) \, dx-\sum_{0\le i< n} \frac h 2 (f(a+ih)+f(a+(i+1)h))\right|\le \frac{K}{n^2} \max_{x\in[a,b]} |f''(x)|,\ \ h=\frac{b-a}{n}, $$ for some $K$ depending only on $a$ and $b$. A bound like this is a rigorous, precise form of the $\approx$.