Squeeze (Sandwich) Theorem for Sequences

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Consider two sequences `x_n` and `y_n`. When we write that `x_n=y_n` we mean that corresponding values are equal, i.e. `x_1=y_1`, `x_2=y_2` etc.

Fact 1. If two sequences `x_n` and `y_n` are equal: `x_n=y_n`, and each of them has limit (finite or infinite): `lim x_n=a` and `lim y_n=b` then `a=b`.

This fact is used in limiting process: from `x_n=y_n` we conclude that `lim x_n=lim y_n`.

Fact 2. If for two sequences `x_n` and `y_n` we have that `x_n>=y_n`, and each of them has limit (finite or infinite): `lim x_n=a` and `lim y_n=b` then `a>=b`.

This fact is used in limiting process: from `x_n>=y_n` we conclude that `lim x_n>=lim y_n`.

Of course sign `>=` can be replaced by sign `<=`, i.e. from `x_n<=y_n` we conclude that `lim x_n<=lim y_n`.

CAUTION! Inequality `x_n>y_n` we can't conclude that `lim x_n>lim y_n`, we can only conclude that `lim x_n>=lim y_n`.

For example, consider two sequences `x_n=1/n` and `y_n=-1/n`. Clearly, `x_n>y_n` but `lim 1/n=lim -1/n=0`.

Fact 3 (Squeeze Theorem for Sequences). Consider three sequences `x_n,y_n,z_n`. If we have that `x_n<=y_n<=z_n`, and sequence `x_n` and `z_n` have same limit (finite or infinite), i.e. `lim x_n=lim z_n=a`, then `lim y_n=a`.

This theorem tells us folowing: if there are three sequences, two of which have same squeeze theorem for sequenceslimit and third is "squeezed" between them, then third will have same limit as first two.

Consider figure to the right. Let sequence drawn in green and sequence drawn in blue have same limit `a`. Since sequence drawn in pink is "squeezed" between them, then it will also have limit `a`.

From this theorem it follows that if for all `n` `a<=y_n<=z_n` and `z_n->a` then `y_n->a`.