Many of you have been making (easily fixable since you have the right ideas) errors in writing up your proofs that involve limits. Basically, you have been writing lim _n&rarr&infin z_n, and assuming that the limit exists, before you actually show the limit exists. It is time to get it down right. Below are some ideas that (hopefully) will help prevent such troubles. The idea behind the fix is to use that the lim-sup and lim-inf of any sequence exist in the extended real numbers R ∪ {+∞, -∞} . So work with the lim-inf & lim-sup (keeping in mind that might be +/- ∞, but atleast they exist) instead of working with lim _n&rarr&infin z_n (which you don't even know if it exists). So read the below. You do not have to quote any of the Lemma/Theorems in your proofs. Just understand the ideas and use the ideas as needed in your proofs.

A Basic Lemma
Let { z_n } be a sequence from R and M ε R. Then the following are equivalent.

• (BL1)   lim _n&rarr&infin z_n exists (in R) and lim _n&rarr&infin z_n = M
• (BL2)   liminf _n&rarr&infin z_n = limsup _n&rarr&infin z_n = M
• (BL3)   For each ε > 0 we have that M - ε < liminf _n&rarr&infin z_n ≤ limsup _n&rarr&infin z_n < M + &epsilon

• -∞ ≤ liminf _n&rarr&infin z_n ≤ limsup _n&rarr&infin z_n ≤ + ∞  
• lim _n&rarr&infin z_n exists (in R, ie. is finite) if only only if liminf _n&rarr&infin z_n = limsup _n&rarr&infin z_n (in R, ie. are finite) ,
  in which case,
  lim _n&rarr&infin z_n = liminf _n&rarr&infin z_n = limsup _n&rarr&infin z_n .

Squeeze Theorem (utility grade)
Let { z_n } be a sequence from R and M ε R.
If there exists squences { a_n } and { b_n } from R and N₀ ε N such that

Squeeze Theorem (working version)
Let { z_n } be a sequence from R and M ε R.
If, for each ε >0 , there exists squences { a_n^ε } and { b_n^ε } from R and N_ε ε N such that