Poor Grade in First Real Analysis Course
A poor grade in a single course (especially in your second year) is unlikely to have a lot of effect on your application most places. But it isn't an indicator of success. When you apply for grad school or a job you need to indicate in the application process that you are an excellent candidate for success going forward. Poor grades don't help you. If managing proofs is really your issue then it will be hard to be a success in mathematics no matter what the field. 

On the other hand, insight into mathematical structures isn't uniform across disciplines. I was great in analysis and topology but marginal (at best) in abstract algebra. I could follow along OK, but had no real insight into the big ideas. 

Analysis can be pretty important in statistics, of course, and probably also in numerical analysis. But much less so in most of CS. 

But, there is tremendous competition for slots in doctoral studies. You will be competing with a lot of people with better grades. That doesn't necessarily make you worse overall, but you need to make the case that you are the one for whom they can easily predict success. Make it so.