How does a geometry-oriented mind learn analysis?

I find it very difficult to understand analysis, because I can't find a way to learn it geometrically. To make my point clearer, let me take calculus as the example in contrast. I find calculus very geometric and I'm very comfortable with it, to name a few famous examples:

1. Newton-Leibniz theorem: a strip under the graph of a (continuous)function, divided by its width, will approach the value of the height as width goes to 0;




2. L'Hospital's Rule: if two functions are approaching 0 at the same point, as we approach that point, the ratio of the heights of the two graphs will be the same as the ratio of the slopes, because $\text{height}\approx \text{slope}\times\Delta x$, and $\Delta x$ is the same for both graphs.

I can keep going, and in fact, most of the important theorems in calculus have such geometric intepretations. I'm aware that there are technicalities that cannot be captureed by the above geometric arguments, but I believe they are still very to the point and I feel I can never forget them.

It seems to be another story for analysis(especially real analysis). Although occasionally there are important content that can be understood geometrically, more often I encounter theorems and concepts that after quite some struggling still no firm geometric intuition can be developed. A lot of times some partial intuition can be developed, but usually trusting these partial intuitions will quickly get myself into errors. I can follow most of the analysis textbooks, i.e. all their logical deductions, proving theorems etc. but the problem is, when I can't form pictures, I can't find positions in my mind for these knowledge, so although I have read a considerable amount of analysis(and done quite some exercises), not much can be "triggered" from my mind when I see a analysis-related problem , say, 4 months after I stop reading an analysis book.

In fact, I post the question here only after reading Poincare's Intuition and Logic in Mathematics, before reading this I assumed everyone thinks in the geometric way so if I can't understand analysis I probably should just try harder. It's quite a headache to me because nothing else(calculus, linear algebra, modern differential geometry etc.) has caused this much of discomfort. Now I'm wondering if I've learnt analysis in a wrong way, any comments or suggestions?