How to approach real analaysis

Question

I'm just starting first year in university in Europe and here there there is no Calculus, instead you jump right into Analysis.

The trouble is, for some time I self-studied through US style books and prepared for uni via How to prove it, OCW and such and just started working through Apostol's Calculus and figured I would eventually progress towards some analysis book.

But, I encountered following curriculum in University course:

1. Logic and Sets
   
2. Functions and Relations
   
3. Real and Complex fields
   
4. Limit of a sequence: intro, convergence, number e,Cauchy's criterion of convergence
   
5. Limit of a function:basic limits, Heine and Cauchy's criterion
   
6. Continuity: intro, Weierstrass, Intermediate value theorem, Cantor's theorems
   
7. Derivative: geometrical interpretation, Fermat, Rolle, Lagrange, Cauchy theorems, Taylor's formula
8. Differential of a function
   
9. Integral
   
10. Sequences
    
11. Metric spaces
    

Now I'm confused. Should I abandon my routine and pick up some analysis books (and if so which one for beginner?) or just continue doing through Apostol and pick up analysis stuff on the go.
Perhaps there is some other route.
(Note that I've taken some calculus in high school but only computational stuff, barely touched theory).
Thanks.