Is $1.0000...$ ( $1$ with infinite zeros) greater than $1.0$? Given that $0.3333...$ is greater than $0.3$ and similarly $0.777...$ is greater than $0.7$, does it follow that the sum of $0.33...$ and $0.77...$ is greater than sum of $0.3$ and $0.7$? 

This is clearer to see when using fraction notation:
$$0.\dot3 = \frac{1}{3},\,0.\dot7 = \frac{7}{9},\,0.3 = \frac{3}{10}, 0.7 = \frac{7}{10}$$
So:
$$0.\dot3 + 0.\dot7 = \frac{1}{3} + \frac{7}{9} = \frac{10}{9} = 1.\dot{1} > 1\\0.3+ 0.7= \frac{3}{10} + \frac{7}{10} = \frac{10}{10} = 1$$
Not only is it greater, it doesn't actually equal $1.000\dots$, but rather $1.111\dots$!

In general, if $a > m$ and $b > n$, then $a+b > m + n$


Note that I've used dots above the number to repeated decimal expansion, for example: $0.\dot1 = 0.111\dots$
