Having Trouble Understanding Meaning Of A Finite-Dimensional Vector Space The definition I have is the following:


  A vector space V is said to be finite-dimensional if there is a finite set of vectors in V that spans V and is said to be infinite-dimensional if no such set exists.


However, with this definition I can't determine whether the vector space $\mathbb{R}^3$ is finite-dimensional or infinite-dimensional (I am assuming that it is finite since the dimension of $\mathbb{R}^3$ is $3$)

Going with my thought process, though, I know that $(1,0,0),(0,1,0),(0,0,1)$ spans $\mathbb{R}^3$. However we can also check that $(2,0,0),(0,2,0),(0,0,2)$ spans $\mathbb{R}^3$. Also note that $(3,0,0),(0,3,0),(0,0,3)$ spans $\mathbb{R}^3$. This process could be continued over and over to show that there are infinitely many vectors that span $\mathbb{R}^3$. 

Wouldn't this mean that $\mathbb{R}^3$ is infinite-dimensional? Because there isn't a finite number of vectors that span $\mathbb{R}^3$. (Again I want to say this isn't the case and that there is something I am overlooking.)
The question asks whether there exists a finite basis of the vector space. If there exists a finite basis, then this vector space is said to be finite dimensional. If not the vector space is infinite dimensional. An example of an infinite dimensional vector space is the vector space of all power series.

Contrast this with the vector space of all polynomials of degree less than or equal to 3, $\mathbb{P}_3 [t]$ which has finite dimension 4 since one basis consists of $\{1,t,t^2,t^3\}$
Definition: A city is said to be  food-friendly if  one can get three different cuisines in a single restaurant.

In London, there is Restaurant A which serves Indian, Chinese and Continental. ANd also Restaurant B that serves Thai, Mexican and Greek food. And restaurant C serving,  Indian, Russian and Japanese.

Now does London meet the above definition of food-friendly city? or do you say there three restaurants A,B, C instead of a single restaurant? So  London is  not food-friendly??