Induction (Homework 1)

Induction (Homework 1) - MATHEMATICAL INDUCTION Suppose we...

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MATHEMATICAL INDUCTION Suppose we want to prove a mathematical statement A ( n ) which is formulated for every natural number n N , i.e. n=1,2,3,4,. ... One very often uses the following method (referred to as “mathematical induc- tion”, or simply “induction”) to verify that the statement is true for all n N . (i) One proves the statement for n = 1 , i.e. A (1) . (ii) One shows that for every n the statement A ( n ) implies the statement A ( n +1) (in other words, for every n it is true that if A ( n ) holds then A ( n + 1) holds too). If we have checked both (i) and (ii) then we know that A (2) is true (namely A (1) is true by (i), and therefore, by (ii) A (2) is also true). But then A (3) is also true (we have just seen that A (2) is true and by (ii) it implies A (3)). Etc. etc. Remark: The induction does not have to begin with n = 1. Suppose we know that A (4) is true and that for all n 4 the statement “ A ( n ) implies A ( n + 1)” holds then we can conclude that A ( n ) is true for all n 4. 1. Examples Example 1.1. Let’s prove for n = 1 , 2 , . . . the statement A ( n ) : n X k =1 k = n ( n + 1) 2 , In english: the sum of the first n natural numbers (beginning with 1) is equal to n ( n +1) 2 . To verify A (1) just note that 1 k =1 = 1 = 1 · (1+1) 2 . Now we need to check that for all n the truth of A ( n ) implies the truth of A ( n + 1). Fix n . We write n +1 X k =1 k = ( n X k =1 k ) + ( n + 1) Since we are assuming the truth of A ( n ) (for our fixed n ) we have n k =1 k = n ( n +1) 2 and we can use this in the last displayed formula. Thus n +1 X k =1 k = n ( n + 1) 2 + ( n + 1) = ( n + 1) ± n 2 + 1 ² = ( n + 1)( n + 2) 2 and this yields A ( n + 1). To summarize we have shown A (1) and for all n we have shown that A ( n ) implies A ( n + 1). Thus, by mathematical induction it follows that the assertion holds for all n . ± 1
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2 Example 1.2. Let’s prove the following statement: For all n N the inequality n < 2 n holds. We label this statement A ( n ). Clearly A (1) holds (namely we have 1 < 2 = 2 1 ). Now let’s fix n and show that the validity of A ( n ) implies the validity of A ( n +1). This follows from n + 1 < 2 n + 1 < 2 n + 2 n = 2 · 2 n = 2 n +1 . where in the first inequality we have used the induction hypothesis (i.e. the assumed truth of A ( n )) and in the second inequality the obvious statement that 1 < 2 n for all n = 1 , 2 , . . . ). By mathematical induction it follows that
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Induction (Homework 1) - MATHEMATICAL INDUCTION Suppose we...

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