Help in understanding a proof in Rudin's Principles of Mathematical analysis Hi I was reading Rudin's Princiaples of MathematicalA Analysis. In the third chapter in tying to prove
If $\{p_n\}$ is a sequence in a metric space $X$ and if $\{p_n\}$ converges $\implies$ $\{p_n\}$ is bounded 
The proof goes as follows
Suppose $p_n \rightarrow p$. There is an integer $N$ such that $\forall n>N \implies d(p_n,p)<1$
Now if we put $r=max\{1,d(p_1,p),d(p_2,p),.....,d(p_N,p)\}$
This implies $d(p_n,p)\leq r  \forall n\in N $
Why can't say $d(p_1,p)$ be infinite which makes r infinite and therefore $\{p_n\}$ unbounded?

The definition of a metric space requires that the distances are real numbers. Infinity is not a real number, so $d(p_1,p)$ cannot be $\infty$.
There are generalizations of metric spaces where we allow infinite distances, but that's a different story.
