Poor Grade in First Real Analysis Course I'm currently a sophomore majoring in pure math with the hopes of going to graduate school to study applied math, specifically numerical analysis, machine learning, statistics. I have A's in my courses such as Linear Algebra, Discrete Math, Calculus, etc. Real Analysis is the first course that is giving me lots of trouble, and I think it is because I was not well-versed in writing proofs coming into the course. I received a B-/C+ on the first exam, and expect to receive a similar grade in the overall course.

How heavily will my grade in this course affect my chances of getting into a good graduate school for applied math (numerical analysis) if I receive A's/A-'s in my all of my progamming (up to an Advanced Algorihms course) and applied classes (PDE, Numerical, Complex Variables)?
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.