Cardinality of the set of functions from N to {0,1,2,3,...,9} is equal to card(R)

Question

I know that the set of infinite sequences on {0,1,2,3,4,5,6,7,8,9} is uncountable, but how to show that it has a bijection to R?

Answer 1

Here's the simple answer if you know enough of the basic cardinality theorems:

$$|\Bbb R|=2^{\aleph_0}\le 10^{\aleph_0}\le (2^{\aleph_0})^{\aleph_0}=2^{\aleph_0\times\aleph_0}=2^{\aleph_0}=|\Bbb R|$$