Converting intuition into mathematics (real analysis). I am taking an introductory course in real analysis. It is very different from most math courses I have taken prior.
So far, everything presented has made intuitive sense to me. But, when it comes time to produce a water-tight proof, my issue is converting intuition into mathematics. I am never sure when a sentence explaining my reasoning could be replaced with mathematics to provide a more rigorous argument.
For example!

Suppose I have a sequence $\{a_{n}\}_{n=1}^{\infty}$ such that $a_{1}=1$ and $a_{n+1}=\frac{1}{6}(a_{n}+2)$.

I'm interested in showing that it has a limit. So, I set off to show that it is monotone and bounded. I've shown that it is monotone decreasing and am confident with how I've done that, so will spare the details. But now, I wish to say something like:

Observe that for all $n\geq1$, $a_{n}>0$. As $\{a_{n}\}$ is monotone decreasing, and bounded below by zero, it must converge.

I'm not convinced this is the best way to say what I want to say. I feel like I'm skipping steps (feels hand-wavey to me).
I've been in this situation a few times so far this semester and thought I'd post to get some insight on turning my intuition into mathematics, in this particular instance and if possible also in general - this is the trouble I'm having. I do not want a full solution to the example above, but I would appreciate the insight mentioned.
Thank you for your time!
 A: Some personal thoughts for the "in general" part:
You are of course not alone with this problem. Many of my fellow students, including myself, had problems in the beginning with turning intuition into a good proof.
Personally, what really helped me when I started, was to always ask myself: 
"What am I really trying to prove. What is the definition of what I am trying to prove.", and then trying to directly solve my problem by sticking strictly to the definitions (for example when trying to prove a series is convergent: $\forall \epsilon \ \exists N \dots$ – working my way through quantifiers) and/or swinging from already proved theorems to lemmas to corollaries etc. until I arrive at the definitions.  
This is also something our professors/tutors always told us in the first semester: If you can visualise something (which is still very possible in first semester calculus), always adjust your intuition to the definitions. Ask yourself what you are trying to prove [as esoterically as it may sound] and stick to the definitions!
...and of course practice makes everything easier with time.
