1) you have to compare inside the loop
2) you have to make sure it's lower, not equal
pointing out some other stuff:
using namespace std;
int num = 0;
//what about the value to hold the current lowest int?
cout << "please enter an integer to determine\n"
<< "the amount of integers I will compare\n"
<< "to find and display the smallest one: ";
cin >> num;
cout << "you may enter your integers now; "; //what if they enter 4531863486816, meaning 4,531,8,6348,68,16
for ( int count = 1; count <= num; count++)
cin >> value;
if ( value == value ) //change and put inside loop
cout << value << " is the smallest integer" << endl;