Even numbers proof help
Assume there is a largest even number, E.
E+2 would also be even, as E must be divisible by 2, so E+2 is divisible by 2, and clearly greater than E.
If E is the largest even number, then −E is the largest negative even number.
−E−2 would also be even, as −E must be divisible by 2, so −E−2 is divisible by 2, and clearly -E - 2 is a greater negative even number than E
Therefore, this contradicts the original assumption, so it must be incorrect.
Therefore, there is an infinite number of even numbers.
Is this proof sufficient, or is something missing? As the textbook's answer was much longer.
EDIT: Textbook's answer is: Suppose that there is a finte number N of even numbers
This finite list can be ordered so that E1 < E2 < E3 < ... < En
Then the largest even number is En
But 2En would also be even and clearly greater than En, so is not in the list.
Therefore, there are more than N even numbers.
This contradicts the initial proposition.
Therefore, there are infinitely many even numbers
E+2 would also be even, as E must be divisible by 2, so E+2 is divisible by 2, and clearly greater than E.
If E is the largest even number, then −E is the largest negative even number.
−E−2 would also be even, as −E must be divisible by 2, so −E−2 is divisible by 2, and clearly -E - 2 is a greater negative even number than E
Therefore, this contradicts the original assumption, so it must be incorrect.
Therefore, there is an infinite number of even numbers.
Is this proof sufficient, or is something missing? As the textbook's answer was much longer.
EDIT: Textbook's answer is: Suppose that there is a finte number N of even numbers
This finite list can be ordered so that E1 < E2 < E3 < ... < En
Then the largest even number is En
But 2En would also be even and clearly greater than En, so is not in the list.
Therefore, there are more than N even numbers.
This contradicts the initial proposition.
Therefore, there are infinitely many even numbers
whats the question?
Original post by TAEuler
Assume there is a largest even number, E.
E+2 would also be even, as E must be divisible by 2, so E+2 is divisible by 2, and clearly greater than E.
If E is the largest even number, then −E is the largest negative even number.
−E−2 would also be even, as −E must be divisible by 2, so −E−2 is divisible by 2, and clearly -E - 2 is a greater negative even number than E
Therefore, this contradicts the original assumption, so it must be incorrect.
Therefore, there is an infinite number of even numbers.
Is this proof sufficient, or is something missing? As the textbook's answer was much longer.
EDIT: Textbook's answer is: Suppose that there is a finte number N of even numbers
This finite list can be ordered so that E1 < E2 < E3 < ... < En
Then the largest even number is En
But 2En would also be even and clearly greater than En, so is not in the list.
Therefore, there are more than N even numbers.
This contradicts the initial proposition.
Therefore, there are infinitely many even numbers
E+2 would also be even, as E must be divisible by 2, so E+2 is divisible by 2, and clearly greater than E.
If E is the largest even number, then −E is the largest negative even number.
−E−2 would also be even, as −E must be divisible by 2, so −E−2 is divisible by 2, and clearly -E - 2 is a greater negative even number than E
Therefore, this contradicts the original assumption, so it must be incorrect.
Therefore, there is an infinite number of even numbers.
Is this proof sufficient, or is something missing? As the textbook's answer was much longer.
EDIT: Textbook's answer is: Suppose that there is a finte number N of even numbers
This finite list can be ordered so that E1 < E2 < E3 < ... < En
Then the largest even number is En
But 2En would also be even and clearly greater than En, so is not in the list.
Therefore, there are more than N even numbers.
This contradicts the initial proposition.
Therefore, there are infinitely many even numbers
The textbook must assume the Es are positive? Otherwise 2En < En
Original post by TAEuler
Assume there is a largest even number, E.
E+2 would also be even, as E must be divisible by 2, so E+2 is divisible by 2, and clearly greater than E.
If E is the largest even number, then −E is the largest negative even number.
−E−2 would also be even, as −E must be divisible by 2, so −E−2 is divisible by 2, and clearly -E - 2 is a greater negative even number than E
Therefore, this contradicts the original assumption, so it must be incorrect.
Therefore, there is an infinite number of even numbers.
Is this proof sufficient, or is something missing? As the textbook's answer was much longer.
EDIT: Textbook's answer is: Suppose that there is a finte number N of even numbers
This finite list can be ordered so that E1 < E2 < E3 < ... < En
Then the largest even number is En
But 2En would also be even and clearly greater than En, so is not in the list.
Therefore, there are more than N even numbers.
This contradicts the initial proposition.
Therefore, there are infinitely many even numbers
E+2 would also be even, as E must be divisible by 2, so E+2 is divisible by 2, and clearly greater than E.
If E is the largest even number, then −E is the largest negative even number.
−E−2 would also be even, as −E must be divisible by 2, so −E−2 is divisible by 2, and clearly -E - 2 is a greater negative even number than E
Therefore, this contradicts the original assumption, so it must be incorrect.
Therefore, there is an infinite number of even numbers.
Is this proof sufficient, or is something missing? As the textbook's answer was much longer.
EDIT: Textbook's answer is: Suppose that there is a finte number N of even numbers
This finite list can be ordered so that E1 < E2 < E3 < ... < En
Then the largest even number is En
But 2En would also be even and clearly greater than En, so is not in the list.
Therefore, there are more than N even numbers.
This contradicts the initial proposition.
Therefore, there are infinitely many even numbers
Textbook's answer doesn't seem longer...?
But yes, your proof is sufficient. You didn't need to dip into the negatives though.
Original post by TAEuler
Assume there is a largest even number, E.
E+2 would also be even, as E must be divisible by 2, so E+2 is divisible by 2, and clearly greater than E.
If E is the largest even number, then −E is the largest negative even number.
−E−2 would also be even, as −E must be divisible by 2, so −E−2 is divisible by 2, and clearly -E - 2 is a greater negative even number than E
Therefore, this contradicts the original assumption, so it must be incorrect.
Therefore, there is an infinite number of even numbers.
Is this proof sufficient, or is something missing? As the textbook's answer was much longer.
EDIT: Textbook's answer is: Suppose that there is a finte number N of even numbers
This finite list can be ordered so that E1 < E2 < E3 < ... < En
Then the largest even number is En
But 2En would also be even and clearly greater than En, so is not in the list.
Therefore, there are more than N even numbers.
This contradicts the initial proposition.
Therefore, there are infinitely many even numbers
E+2 would also be even, as E must be divisible by 2, so E+2 is divisible by 2, and clearly greater than E.
If E is the largest even number, then −E is the largest negative even number.
−E−2 would also be even, as −E must be divisible by 2, so −E−2 is divisible by 2, and clearly -E - 2 is a greater negative even number than E
Therefore, this contradicts the original assumption, so it must be incorrect.
Therefore, there is an infinite number of even numbers.
Is this proof sufficient, or is something missing? As the textbook's answer was much longer.
EDIT: Textbook's answer is: Suppose that there is a finte number N of even numbers
This finite list can be ordered so that E1 < E2 < E3 < ... < En
Then the largest even number is En
But 2En would also be even and clearly greater than En, so is not in the list.
Therefore, there are more than N even numbers.
This contradicts the initial proposition.
Therefore, there are infinitely many even numbers
At this level (i.e. for a problem this simple), I think you do need to make some kind of comment about "if there's a finite number of even numbers, then there must be a largest even number".
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