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Limit on the Terms of a Fibonacci Sequence

The Fibonacci sequence is defined as:
\[a_1=1, \: a_2=1, a_{n+2}=a_{n+1}+a_n\]

The first few terms of the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34
Though the sequence increases without limit, each term  
\[a_=n\]
  is less than  
\[2^n\]
. We can prove this with induction.
Let  
\[P(k)\]
  be the statement "
\[a_k \lt 2^k\]
".
\[a_1 =1 \lt 2^1=2\]
  so  
\[P(1)\]
  is true.
Suppose now that  
\[P(k)\]
  is true for  
\[n=1,2,...,k\]
.
Then  
\[a_{k+2}=a_{k+1}+a_k \lt 2^{k+1}+2^k=2n (2+1) \lt 2^k (2+2)=2^k \times 2^2=2^{k+2}\]
.
Hence  
\[P(k+1}\]
  is true.

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