<p>Weasel8488, I am not sure if your post was addressed to me, but just to clarify, I am a current math major who considers going to grad school.</p>
<p>The section I quoted initially struck me because I didn’t experience real analysis to be that difficult, and at the same time I am aware that math classes at my college are less rigorous than math classes at Princeton. Wait, I should rephrase that. Our math classes are rigorous but less intense/difficult than math classes at most equally or more selective colleges. </p>
<p>I am having a really hard time imagining what a more intense math course, or a more intense (grad) program, would feel like, and the section from the graduate school guide I quoted slightly scared me.</p>
<p>Thanks for all of your responses, even the ones that went off-topic. To contribute to the discussion above, I believe talent is mostly acquired through practice with a little bit of inborn talent/luck. I came to that conviction after spending a summer implementing a neural network simulation program and trying to teach artificial networks different tasks. We initialized the networks with a small degree of randomness, and most networks needed about the same amount of time to learn each task. But in each experiment there were a few outliers - networks that never managed to learn the given task, or that outperformed all other networks (e.g. only needed 100 training steps instead of 1,000).</p>